The large Witt graph, also called the octad graph (Brouwer) or Witt graph (DistanceRegular.org), is the graph whose vertices are the 759 blocks of a Steiner system in which two blocks are adjacent whenever they are disjoint (Brouwer et al. 1989, p. 366).
Perhaps the simplest construction is by selecting the 759 codewords of weight 8 of the extended binary Golay code and joining two words when they have disjoint support (i.e., if the codeword vectors are orthogonal).
It is a distance-regular graph with intersection array and is also distance-transitive. It is an integral graph with graph spectrum . Its automorphism group has order , where is the largest Mathieu group. Its chromatic number is apparently unknown.
The large Witt graph is implemented in the Wolfram Language as GraphData["LargeWittGraph"].