Editing Leech lattice (section)

==Symmetries==

The Leech lattice is highly symmetrical.  Its [[automorphism group]] is the [[Conway group]] Co<sub>0</sub>, which is of order  8&nbsp;315&nbsp;553&nbsp;613&nbsp;086&nbsp;720&nbsp;000.  The center of Co<sub>0</sub> has two elements, and the quotient of Co<sub>0</sub> by this center is the Conway group Co<sub>1</sub>, a finite simple group.  Many other [[sporadic group]]s, 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 hyperplanes of reflection symmetry. In other words, the Leech lattice is [[chiral]]. It also has far fewer symmetries than the 24-dimensional hypercube and simplex, or even the Cartesian product of three copies of the [[E8 lattice]].

The automorphism group was first described by [[John Horton Conway|John Conway]]. The 398034000 vectors of norm 8 fall into 8292375 'crosses' of 48 vectors. Each cross contains 24 mutually orthogonal vectors and their negatives, 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 [[Binary Golay code|Golay code]], namely 2<sup>12</sup> &times; |M<sub>24</sub>|. Hence the full automorphism group of the Leech lattice has order 8292375 &times; 4096 &times; 244823040, or 8&nbsp;315&nbsp;553&nbsp;613&nbsp;086&nbsp;720&nbsp;000.