Maximal cliques in the graph of 5-ary simplex codes of dimension two

We consider the induced subgraph of the corresponding Grassmann graph formed by q-ary simplex codes of dimension 2, q≥ 5. This graph contains precisely two types of maximal cliques. If q=5, then for any two maximal cliques of the same type there is a monomial linear automorphism transferring one of them to the other. Examples concerning the cases q=7,11 finish the note.