The automorphism group of a class of strongly regular graphs related to Q(6, q)

In [A. Devillers, H. Van Maldeghem, Partial linear spaces built on hexagons, European J. Combin. 28 (2007) 901-915], Devillers and Van Maldeghem determined the automorphism group of four classes of geometries that have as collinearity graph the graph Gamma(q) of all elliptic hyperplanes of a given parabolic quadric Q(6, q) in PG(6, q) (adjacency is given by intersecting in a tangent 4-space). In their introduction they mention that at the time they were not able to determine the full automorphism group of Gamma(q), but that their results might be useful for proving that it is isomorphic to Pro(7, q). In this note we use one of their results to prove that this is indeed the case