Symmetries
The Leech lattice is highly symmetrical. Its automorphism group is the double cover of the Conway group Co1; its order is 8 315 553 613 086 720 000. Many other sporadic simple groups, such as the remaining Conway groups and Mathieu groups, can be constructed as the stabilizers of various configurations of vectors in the Leech lattice.
Despite having such a high rotational symmetry group, the Leech lattice does not possess any lines of reflection symmetry. In other words, the Leech lattice is chiral.
The automorphism group was first described by John Conway. The 398034000 vectors of norm 8 fall into 8292375 'crosses' of 48 vectors. Each cross contains 24 mutually orthogonal vectors and their inverses, and thus describe the vertices of a 24-dimensional orthoplex. Each of these crosses can be taken to be the coordinate system of the lattice, and has the same symmetry of the Golay code, namely 212 × |M24|. Hence the full automorphism group of the Leech lattice has order 8292375 × 4096 × 244823040, or 8 315 553 613 086 720 000.
Read more about this topic: Leech Lattice