The Foster graph is one of the 13 such graphs. It is the unique distance-transitive graph with intersection array {3,2,2,2,2,1,1,1;1,1,1,1,2,2,2,3}.
The Foster graph falls under a category of 13 such graphs. Specifically, it is the only distance-transitive graph with a particular arrangement of vertex degrees: {3,2,2,2,2,1,1,1;1,1,1,1,2,2,2,3}.