the 1 isometry classes of irreducible [13,8,4]_2 codes are:

code no       1:
================
1 1 1 1 1 1 0 0 0 0 0 0 0
1 1 1 0 0 0 1 0 0 0 0 0 0
1 1 0 1 0 0 0 1 0 0 0 0 0
1 0 1 1 0 0 0 0 1 0 0 0 0
0 1 1 1 0 0 0 0 0 1 0 0 0
1 1 0 0 1 0 0 0 0 0 1 0 0
1 0 1 0 1 0 0 0 0 0 0 1 0
0 1 1 0 1 0 0 0 0 0 0 0 1
the automorphism group has order 576
and is strongly generated by the following 7 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 1 0 1 0 
1 1 0 0 1 
, 
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 1 0 
1 0 1 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 1 0 0 
0 0 0 1 0 
1 1 0 0 1 
, 
1 0 0 0 0 
0 0 1 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 0 0 1 
, 
1 0 1 1 0 
1 1 0 1 0 
0 0 0 1 0 
0 0 1 0 0 
1 0 1 0 1 
, 
1 0 1 0 1 
0 1 1 0 1 
1 1 0 0 1 
1 1 0 1 0 
0 0 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(4, 8)(5, 11)(9, 10)(12, 13), 
(4, 11)(5, 8)(9, 13)(10, 12), 
(4, 9)(5, 12)(8, 10)(11, 13), 
(3, 7)(5, 11)(9, 10), 
(2, 3)(8, 9)(11, 12), 
(1, 9)(2, 8)(3, 4)(5, 12)(7, 10), 
(1, 13, 2, 12)(3, 5, 7, 11)(4, 8)
orbits: { 1, 9, 12, 10, 13, 4, 8, 5, 11, 2, 7, 3 }, { 6 }