the 1 isometry classes of irreducible [6,2,5]_5 codes are:

code no       1:
================
1 1 1 1 4 0
4 3 2 1 0 4
the automorphism group has order 120
and is strongly generated by the following 8 elements:
(
2 0 0 0 
0 3 0 0 
4 4 4 4 
4 3 2 1 
, 
1 0 0 0 
0 3 0 0 
0 0 0 2 
1 1 1 1 
, 
4 0 0 0 
1 1 1 1 
0 0 0 4 
0 0 4 0 
, 
2 0 0 0 
0 0 0 2 
0 2 0 0 
3 3 3 3 
, 
4 0 0 0 
3 1 4 2 
0 1 0 0 
0 0 3 0 
, 
2 2 2 2 
0 0 3 0 
3 0 0 0 
0 0 0 3 
, 
0 0 0 1 
4 4 4 4 
1 0 0 0 
0 1 0 0 
, 
2 4 1 3 
0 0 4 0 
0 0 0 2 
3 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(3, 5)(4, 6), 
(3, 6, 5, 4), 
(2, 5)(3, 4), 
(2, 3, 5, 4), 
(2, 3, 4, 6), 
(1, 3, 2, 5), 
(1, 3, 5, 2, 4), 
(1, 4, 3, 2, 5, 6)
orbits: { 1, 5, 4, 6, 3, 2 }