It is the largest of the three sporadic Conway groups and can be obtained as the quotient of Co0 (group of automorphisms of the Leech lattice Λ that fix the origin) by its center, which consists of the scalar matrices ±1.
It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II25,1.
Some rather cryptic comments in Witt's collected works suggest that he found the Leech lattice and possibly the order of its automorphism group in unpublished work in 1940.
The outer automorphism group is trivial and the Schur multiplier has order 2.
The smallest faithful permutation representation of Co1 is on the 98280 pairs {v,–v} of norm 4 vectors.