[1] The Gosset graph can be explicitly constructed as follows: the 56 vertices are the vectors in R8 obtained by permuting the coordinates and possibly taking the opposite of the vector (3, 3, −1, −1, −1, −1, −1, −1).

An alternative construction is based on the 8-vertex complete graph K8.

The vertices of the Gosset graph can be identified with two copies of the set of edges of K8.

Two vertices of the Gosset graph that come from different copies are adjacent if they correspond to disjoint edges of K8; two vertices that come from the same copy are adjacent if they correspond to edges that share a single vertex.

In the vector representation of the Gosset graph, two vertices are at distance two when their inner product is −8 and at distance three when their inner product is −24 (which is only possible if the vectors are each other's opposite).