[2] It was first constructed by Mesner (1956)[3] and rediscovered in 1968 by Donald G. Higman and Charles C. Sims as a way to define the Higman–Sims group, a subgroup of index two in the group of automorphisms of the Hoffman–Singleton graph.

Take all 70 possible 4-sets of vertices, and retain only the ones whose XOR evaluates to 000; there are 14 such 4-sets, corresponding to the 6 faces + 6 diagonal-rectangles + 2 parity tetrahedra.

There are then 30 different ways to relabel the vertices (i.e., 30 different designs that are all isomorphic to each other by permutation of the points).

Connect rows to each other if they have exactly one element in common (there are 4x4 = 16 such neighbors).

[6] The outer elements induce odd permutations on the graph.

As mentioned above, there are 352 ways to partition the Higman–Sims graph into a pair of Hoffman–Singleton graphs; these partitions actually come in 2 orbits of size 176 each, and the outer elements of the Higman–Sims group swap these orbits.

The Higman–Sims graph naturally occurs inside the Leech lattice: if X, Y and Z are three points in the Leech lattice such that the distances XY, XZ and YZ are

Furthermore, the set of all automorphisms of the Leech lattice (that is, Euclidean congruences fixing it) which fix each of X, Y and Z is the Higman–Sims group (if we allow exchanging X and Y, the order 2 extension of all graph automorphisms is obtained).