the 2 isometry classes of irreducible [28,23,3]_2 codes are:

code no       1:
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1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 2304
and is strongly generated by the following 10 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 1 0 0 1 
1 1 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 0 1 0 1 
1 0 1 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 1 1 0 1 
0 1 1 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 0 0 1 0 
1 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 1 1 0 0 
1 1 0 1 0 
1 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 1 0 1 1 
1 0 0 1 0 
1 1 1 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
1 0 0 1 1 
0 1 1 1 0 
1 1 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 0 0 1 
0 1 0 1 1 
1 0 1 0 1 
, 
1 0 0 0 0 
1 1 0 0 0 
0 0 1 0 0 
1 0 0 0 1 
0 0 0 1 0 
, 
0 1 0 0 0 
1 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
0 1 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(4, 20)(5, 13)(11, 19)(12, 18)(14, 24)(15, 23)(16, 22)(17, 21), 
(4, 22)(5, 15)(11, 21)(12, 24)(13, 23)(14, 18)(16, 20)(17, 19), 
(4, 23)(5, 16)(11, 24)(12, 21)(13, 22)(14, 19)(15, 20)(17, 18), 
(4, 11)(5, 18)(12, 13)(14, 15)(16, 17)(19, 20)(21, 22)(23, 24), 
(3, 9)(4, 13)(5, 18)(8, 10)(11, 12)(14, 15)(16, 17)(19, 20)(21, 24)(22, 23)(25, 27)(26, 28), 
(3, 27)(4, 11)(5, 17)(8, 28)(9, 25)(10, 26)(12, 13)(14, 20)(15, 19)(16, 18)(21, 22)(23, 24), 
(3, 26)(4, 20, 22, 16)(5, 23, 15, 13)(8, 25)(9, 28)(10, 27)(11, 19, 21, 17)(12, 18, 24, 14), 
(3, 23, 20)(4, 15, 27)(5, 10, 22)(8, 24, 19)(9, 21, 18)(11, 14, 28)(12, 17, 25)(13, 16, 26), 
(2, 7)(4, 5, 11, 18)(9, 10)(12, 20, 13, 19)(14, 21, 15, 22)(16, 24, 17, 23)(25, 26), 
(1, 7, 2)(5, 18, 19)(8, 10, 9)(11, 13, 12)(15, 17, 16)(21, 22, 23)(25, 26, 27)
orbits: { 1, 2, 7 }, { 3, 9, 27, 26, 20, 25, 28, 18, 10, 15, 16, 4, 19, 14, 23, 12, 8, 17, 5, 21, 11, 22, 13, 24 }, { 6 }

code no       2:
================
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
the automorphism group has order 64512
and is strongly generated by the following 12 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 1 1 1 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 0 1 
0 0 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 0 0 1 
0 0 0 1 0 
1 1 1 0 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 1 0 0 1 
0 1 1 1 1 
0 1 1 0 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 1 0 0 
0 0 1 1 1 
, 
1 0 0 0 0 
0 1 0 1 1 
1 1 0 0 1 
0 0 0 0 1 
1 1 1 0 0 
, 
1 0 0 0 0 
1 0 1 1 1 
0 0 1 1 0 
0 0 0 1 0 
1 1 1 0 0 
, 
1 0 0 0 0 
1 1 0 0 1 
0 1 0 1 1 
0 0 0 0 1 
1 1 1 0 0 
, 
1 0 0 0 0 
0 0 0 0 1 
0 0 1 1 0 
0 1 0 1 0 
1 0 1 0 0 
, 
0 0 0 0 1 
1 1 0 0 1 
0 1 0 0 0 
0 1 1 0 1 
1 0 0 1 1 
, 
0 1 0 0 0 
1 0 0 0 0 
0 0 0 0 1 
0 0 1 0 0 
1 1 0 1 0 
, 
0 1 1 1 1 
1 0 1 0 1 
1 1 0 1 1 
1 1 1 1 1 
1 1 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 6)(17, 28)(18, 27)(19, 26)(20, 25)(21, 24)(22, 23), 
(4, 5)(11, 17)(12, 18)(13, 19)(14, 20)(15, 21)(16, 22), 
(3, 19)(5, 10)(8, 18)(9, 17)(14, 25)(15, 24)(16, 23), 
(3, 18)(4, 28)(5, 9)(6, 11)(8, 19)(10, 17)(12, 26)(13, 27), 
(3, 4)(5, 26)(6, 19)(8, 11)(9, 12)(10, 13)(17, 27)(18, 28), 
(2, 15, 24)(3, 26, 19)(4, 10, 5)(7, 14, 25)(8, 27, 18)(9, 17, 11)(16, 22, 23), 
(2, 27)(3, 14)(5, 10)(7, 26)(8, 15)(9, 17)(12, 21)(13, 20)(16, 23)(18, 24)(19, 25), 
(2, 13, 15, 26, 21, 19)(3, 24)(4, 10, 5)(7, 12, 14, 27, 20, 18)(8, 25)(9, 17, 11)(16, 22, 23), 
(2, 27, 10, 15, 8, 5)(3, 17, 7, 26, 9, 14)(4, 21, 18, 6, 24, 12)(11, 20, 19, 28, 25, 13)(16, 23, 22), 
(1, 10, 16, 6, 23, 5)(2, 3, 15, 26, 24, 19)(4, 22)(8, 12, 18)(9, 11, 17)(13, 21)(14, 25, 20), 
(1, 2)(3, 4, 19, 10, 13, 5)(6, 26)(8, 12, 18)(9, 11, 17)(14, 25, 20)(15, 23, 21, 16, 24, 22), 
(1, 11, 22, 28)(2, 21)(3, 14, 19, 25)(4, 6)(5, 10)(7, 26, 20, 13)(9, 23, 17, 16)(15, 24)
orbits: { 1, 5, 2, 28, 6, 4, 10, 9, 26, 8, 23, 13, 24, 27, 19, 21, 17, 18, 22, 11, 16, 3, 12, 7, 15, 25, 20, 14 }