Leech Lattice

A 24-D Euclidean lattice. An Automorphism of the Leech lattice modulo a center of two leads to the Conway Group ${\it Co}_1$. Stabilization of the 1- and 2-D sublattices leads to the Conway Groups ${\it Co}_2$ and ${\it Co}_3$, the Higman-Sims Group HS and the McLaughlin Group McL.

The Leech lattice appears to be the densest Hypersphere Packing in 24-D, and results in each Hypersphere touching 195,560 others.

See also Barnes-Wall Lattice, Conway Groups, Coxeter-Todd Lattice, Higman-Sims Group, Hypersphere, Hypersphere Packing, Kissing Number, McLaughlin Group

References

Conway, J. H. and Sloane, N. J. A. ``The 24-Dimensional Leech Lattice $\Lambda_{24}$,'' ``A Characterization of the Leech Lattice,'' ``The Covering Radius of the Leech Lattice,'' ``Twenty-Three Constructions for the Leech Lattice,'' ``The Cellular of the Leech Lattice,'' ``Lorentzian Forms for the Leech Lattice.'' §4.11, Ch. 12, and Chs. 23-26 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 131-135, 331-336, and 478-526, 1993.

Leech, J. ``Notes on Sphere Packings.'' Canad. J. Math. 19, 251-267, 1967.

Wilson, R. A. ``Vector Stabilizers and Subgroups of Leech Lattice Groups.'' J. Algebra 127, 387-408, 1989.