Is there a McLaughlin geometry?

It has long been an open problem whether or not there exists a partial geometry with parameters (s,t,α)=(4,27,2). Such a partial geometry, which we call a McLaughlin geometry, would have the McLaughlin graph as point graph. In this note we use tools from computational group theory and computational graph theory to show that a McLaughlin geometry cannot have certain automorphisms, nor can such a geometry satisfy the Axiom of Pasch.