- It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1.
- It is even; i.e., the square of the length of any vector in Λ is an even integer.
- The shortest length of any non-zero vector in Λ is 2.

The Leech lattice can be explicitly constructed as the set of vectors of the form 2^{−3/2}(*a*_{1}, *a*_{2}, ..., *a*_{24}) where the *a _{i}* are integers such that

and the set of coordinates

The Leech lattice is highly symmetrical. Its automorphism group is the double cover of the Conway group Co_{1}; its order is approximately 8.3(10)^{18}.

See:

- Conway, J. H.; Sloane, N. J. A. (1999).
*Sphere packings, lattices and groups.*(3rd ed.) With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften, 290. New York: Springer-Verlag. ISBN 0-387-98585-9.