the 1 isometry classes of irreducible [6,1,6]_2 codes are:

code no       1:
================
1 1 1 1 1 1
the automorphism group has order 720
and is strongly generated by the following 7 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 
0 0 0 0 1 
0 0 1 0 0 
0 0 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 1 1 1 
0 0 1 0 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 0 0 1 0 
0 0 1 0 0 
0 1 0 0 0 
0 0 0 0 1 
, 
0 0 0 0 1 
0 0 1 0 0 
0 1 0 0 0 
1 0 0 0 0 
0 0 0 1 0 
, 
1 1 1 1 1 
1 0 0 0 0 
0 0 0 1 0 
0 1 0 0 0 
0 0 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(5, 6), 
(4, 5), 
(3, 4, 5), 
(3, 4, 6), 
(2, 4), 
(1, 4, 5)(2, 3), 
(1, 2, 4, 3, 5, 6)
orbits: { 1, 5, 6, 4, 3, 2 }