  We consider the induced subgraph of the corresponding Grassmann graph formed
by $q$-ary simplex codes of dimension $2$, $q\ge 5$. 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.
