The automorphism group 
of the Leech lattice modulo
a center of order two is called "the" Conway group. There are 15 exceptional
conjugacy classes of the Conway group. This group,
combined with the groups 
and 
obtained similarly from the Leech
lattice by stabilization of the one- and two-dimensional sublattices, are collectively
called Conway groups.
The Conway groups are sporadic groups. The are implemented in the Wolfram Language as ConwayGroupCo1[], ConwayGroupCo2[], and ConwayGroupCo3[].
The following table summarizes some properties of the Conway groups, where 
indicates the transitivity and 
is the length of the minimal permutation support.
| group |  |  | order | factorization |
 | 2 | 276 | 495766656000 |  |
 | 1 | 2300 | 42305421312000 |  |
 | 1 | 98280 | 4157776806543360000 |  |
