The graphs of Hoffman-Singleton, Higman-Sims, and McLaughlin, and the Hermitian curve of degree 6 in characteristic 5

We construct the graphs of Hoffman-Singleton, Higman-Sims, and McLaughlin from certain relations on the set of non-singular conics totally tangent to the Hermitian curve of degree 6 in characteristic 5. We then interpret this geometric construction in terms of the subgroup structure of the automorphism group of this Hermitian curve.
Comment: 18pages