The Monster group is the largest, most fascinating, and most mysterious of the so-called sporadic groups. It was constructed by Robert Griess at Princeton in 1982, having been predicted to exist by him and Bernd Fischer in 1973, and was named the Monster by John Conway.
Think of the Monster group as a preposterous snowflake with more than 1,050 symmetries that exists in a space of 196,883 dimensions. It contains the following number of elements: 246 × 320 × 59 × 76 × 112 × 133 × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71 = 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 ~ 8 × 1053 (more than the number of quarks in the Sun).
Despite these impressive credentials, however, it is still classified as a simple group, meaning that it doesn't have any normal subgroups other than the identity element and itself. All 26 simple groups have now been classified and the Monster is far and away the biggest.
At first, it seemed that the Monster was just a curiosity – a Guinness Book record of pure math. Its only "useful" application seemed to be to give the best way for packing spheres in 24 dimensions! In ordinary three-dimensional space (and also four and five dimensions), the grocer's way of stacking oranges in a hexagonal lattice is thought to be the tightest possible (see Kepler's conjecture). But as the number of dimensions increases, the optimal packing method changes. A 24-dimensional grocer would get the most efficient arrangement of his 24-dimensional oranges by using the same symmetry as that of the Monster. This is unlikely to be immediately useful. Much more interesting, however, is the connection that has been found between the symmetry of the Monster and one of the most promising unifying theories in physics – string theory – which has been revealed by the Monstrous Moonshine conjecture.
1. Conway, J. H. and Sloane, N. J. A. "The Monster Group and its 196884-Dimensional Space" and "A Monster Lie Algebra?" Chs. 29-30 in Sphere Packings Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 554–571, 1993.