It has several unusual properties, arising from Conway's discovery that it has a norm zero Weyl vector.
In particular it is closely related to the Leech lattice Λ, and has the Conway group Co1 at the top of its automorphism group.
Write Rm,n for the m+n-dimensional vector space Rm+n with the inner product of (a1,...,am+n) and (b1,...,bm+n) given by The lattice II25,1 is given by all vectors (a1,...,a26) in R25,1 such that either all the ai are integers or they are all integers plus 1/2, and their sum is even.
and we recover one of the definitions of Λ. Conway showed that the roots (norm 2 vectors) having inner product –1 with w=(0,0,1) are the simple roots of the reflection group.
In other words, the simple roots can be identified with the points of the Leech lattice, and moreover this is an isometry from the set of simple roots to the Leech lattice.
There are 24 orbits of primitive norm 0 vectors, corresponding to the 24 Niemeier lattices.
There are 121 orbits of vectors v of norm –2, corresponding to the 121 isomorphism classes of 25-dimensional even lattices L of determinant 2.
In this correspondence, the lattice L is isomorphic to the orthogonal complement of the vector v. There are 665 orbits of vectors v of norm –4, corresponding to the 665 isomorphism classes of 25-dimensional unimodular lattices L. In this correspondence, the index 2 sublattice of the even vectors of the lattice L is isomorphic to the orthogonal complement of the vector v. There are similar but increasingly complicated descriptions of the vectors of norm –2n for n=3, 4, 5, ..., and the number of orbits of such vectors increases quite rapidly.