Leech Lattice
A 24-D Euclidean lattice. An Automorphism of the Leech lattice modulo a center of two leads to theConway Group 
. Stabilization of the 1- and 2-D sublattices leads to theConway Groups 
and 
, the Higman-Sims Group HS and the McLaughlin GroupMcL.
The Leech lattice appears to be the densest Hypersphere Packing in 24-D, and results in each Hyperspheretouching 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 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.
,'' ``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.
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