the 6 isometry classes of irreducible [15,7,5]_2 codes are:

code no       1:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
1 0 1 0 1 0 1 0 0 0 0 1 0 0 0
0 1 0 1 0 1 1 0 0 0 0 0 1 0 0
0 1 1 0 1 0 0 1 0 0 0 0 0 1 0
0 0 1 1 0 1 0 1 0 0 0 0 0 0 1
the automorphism group has order 2
and is strongly generated by the following 1 elements:
(
0 0 0 0 0 0 1 0 
0 1 0 1 0 1 1 0 
1 0 1 0 1 0 1 0 
1 1 1 1 1 1 1 1 
0 0 0 0 1 0 0 0 
0 1 1 0 1 0 0 1 
1 0 0 0 0 0 0 0 
1 1 1 1 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(1, 7)(2, 13)(3, 12)(4, 9)(6, 14)(8, 10)(11, 15)
orbits: { 1, 7 }, { 2, 13 }, { 3, 12 }, { 4, 9 }, { 5 }, { 6, 14 }, { 8, 10 }, { 11, 15 }

code no       2:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
1 0 1 0 1 0 1 0 0 0 0 1 0 0 0
0 1 0 1 0 1 1 0 0 0 0 0 1 0 0
0 1 1 0 1 0 0 1 0 0 0 0 0 1 0
0 0 0 1 1 0 1 1 0 0 0 0 0 0 1
the automorphism group has order 2
and is strongly generated by the following 1 elements:
(
0 0 1 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 0 1 0 
0 1 0 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 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(1, 3)(2, 5)(4, 7)(6, 8)(10, 12)(11, 14)(13, 15)
orbits: { 1, 3 }, { 2, 5 }, { 4, 7 }, { 6, 8 }, { 9 }, { 10, 12 }, { 11, 14 }, { 13, 15 }

code no       3:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
1 0 1 0 1 0 1 0 0 0 0 1 0 0 0
0 1 0 1 0 1 1 0 0 0 0 0 1 0 0
1 1 0 1 1 0 0 1 0 0 0 0 0 1 0
1 0 0 1 0 0 1 1 0 0 0 0 0 0 1
the automorphism group has order 4
and is strongly generated by the following 2 elements:
(
1 0 0 0 0 0 0 0 
1 0 0 1 0 0 1 1 
0 0 1 0 0 0 0 0 
1 1 0 0 1 1 0 0 
0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 1 
0 0 0 0 1 0 0 0 
0 0 0 0 0 1 0 0 
, 
0 1 0 1 0 1 1 0 
0 0 0 0 0 1 0 0 
1 0 1 0 1 0 1 0 
0 0 0 0 1 0 0 0 
0 0 0 1 0 0 0 0 
0 1 0 0 0 0 0 0 
1 1 0 0 1 1 0 0 
1 0 0 1 0 0 1 1 
)
acting on the columns of the generator matrix as follows (in order):
(2, 15)(4, 11)(5, 7)(6, 8)(9, 10), 
(1, 13)(2, 6)(3, 12)(4, 5)(7, 11)(8, 15)
orbits: { 1, 13 }, { 2, 15, 6, 8 }, { 3, 12 }, { 4, 11, 5, 7 }, { 9, 10 }, { 14 }

code no       4:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
1 0 1 0 1 0 1 0 0 0 0 1 0 0 0
0 1 0 1 0 1 1 0 0 0 0 0 1 0 0
1 1 0 1 1 0 0 1 0 0 0 0 0 1 0
0 0 0 1 1 0 1 1 0 0 0 0 0 0 1
the automorphism group has order 24
and is strongly generated by the following 3 elements:
(
1 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 
0 0 0 0 0 1 0 0 
0 0 1 0 0 0 0 0 
0 0 0 1 0 0 0 0 
0 0 0 0 0 0 1 0 
1 1 1 1 1 1 1 1 
, 
0 1 0 0 0 0 0 0 
1 0 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 
0 0 0 0 1 0 0 0 
0 0 0 1 0 0 0 0 
0 0 1 0 0 0 0 0 
0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 1 
, 
1 1 0 1 1 0 0 1 
1 1 1 1 1 1 1 1 
1 0 0 0 0 0 0 0 
0 1 0 1 0 1 1 0 
0 1 0 0 0 0 0 0 
1 0 1 0 1 0 1 0 
0 0 0 0 0 0 1 0 
0 0 0 1 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(3, 5)(4, 6)(8, 9)(10, 11)(14, 15), 
(1, 2)(3, 6)(4, 5)(10, 11)(12, 13), 
(1, 3, 14)(2, 5, 9)(4, 8, 13)(6, 15, 12)
orbits: { 1, 2, 14, 9, 15, 3, 8, 5, 6, 4, 12, 13 }, { 7 }, { 10, 11 }

code no       5:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
1 0 1 0 1 0 1 0 0 0 0 1 0 0 0
0 1 0 1 0 1 1 0 0 0 0 0 1 0 0
0 1 1 1 1 0 0 1 0 0 0 0 0 1 0
1 0 0 1 1 1 0 1 0 0 0 0 0 0 1
the automorphism group has order 24
and is strongly generated by the following 3 elements:
(
1 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 1 1 1 1 1 1 
0 0 0 0 1 0 0 0 
0 0 1 0 0 0 0 0 
0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 
, 
0 1 0 1 0 1 1 0 
0 0 0 1 0 0 0 0 
0 0 0 0 0 1 0 0 
0 1 0 0 0 0 0 0 
1 0 1 0 1 0 1 0 
0 0 1 0 0 0 0 0 
1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 1 
, 
0 0 0 0 0 0 1 0 
1 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 
1 1 1 1 1 1 1 1 
0 0 0 0 0 1 0 0 
0 1 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(2, 7)(3, 6)(4, 9)(10, 14)(11, 12)(13, 15), 
(1, 13)(2, 4)(3, 6)(5, 12)(7, 10)(9, 14), 
(1, 2, 7)(4, 9, 5)(10, 15, 12)(11, 13, 14)
orbits: { 1, 13, 7, 15, 11, 2, 10, 12, 14, 4, 5, 9 }, { 3, 6 }, { 8 }

code no       6:
================
1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
0 0 1 1 1 0 1 0 0 0 0 1 0 0 0
1 0 1 0 1 1 1 0 0 0 0 0 1 0 0
1 0 1 1 0 1 0 1 0 0 0 0 0 1 0
0 0 0 1 1 1 0 1 0 0 0 0 0 0 1
the automorphism group has order 360
and is strongly generated by the following 5 elements:
(
1 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 
0 0 0 0 1 0 0 0 
0 0 0 1 0 0 0 0 
0 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 1 
0 0 0 0 0 0 1 0 
, 
1 0 0 0 0 0 0 0 
0 0 1 1 1 0 1 0 
1 0 1 0 1 1 1 0 
0 0 0 1 0 0 0 0 
1 1 1 1 1 1 1 1 
1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 1 
0 0 0 0 0 0 1 0 
, 
1 0 0 0 0 0 0 0 
1 1 0 0 1 1 0 0 
1 0 1 0 1 1 1 0 
0 0 0 1 1 1 0 1 
0 0 0 0 0 1 0 0 
0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 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 
1 1 0 0 1 1 0 0 
0 0 0 0 0 0 1 0 
1 1 1 1 0 0 0 0 
0 0 0 1 0 0 0 0 
0 0 0 0 0 1 0 0 
1 1 1 1 1 1 1 1 
, 
0 0 0 1 1 1 0 1 
1 1 0 0 1 1 0 0 
0 0 1 0 0 0 0 0 
1 1 1 1 0 0 0 0 
1 0 0 0 0 0 0 0 
1 1 1 1 1 1 1 1 
0 1 0 0 0 0 0 0 
0 0 1 1 1 0 1 0 
)
acting on the columns of the generator matrix as follows (in order):
(3, 6)(4, 5)(7, 8)(10, 11)(12, 15)(13, 14), 
(2, 12)(3, 13)(5, 9)(6, 10)(7, 8)(11, 14), 
(2, 11)(3, 13)(4, 15)(5, 6)(9, 10)(12, 14), 
(1, 11, 3)(4, 6, 7)(5, 13, 10)(8, 14, 9), 
(1, 5, 14, 15)(2, 7, 13, 11)(4, 12, 8, 10)(6, 9)
orbits: { 1, 3, 15, 6, 13, 11, 12, 4, 14, 10, 5, 9, 7, 2, 8 }