Bases for quasisimple linear groups

Let V be a vector space of dimension d over F _q , a finite field of q elements, and let G ≤ GL(V) ≅≠ GL _d (q) be a linear group. A base for G is a set of vectors whose pointwise stabilizer in G is trivial. We prove that if G is a quasisimple group (i.e., G is perfect and G/Z(G) is simple) acting irreducibly on V, then excluding two natural families, G has a base of size at most 6. The two families consist of alternating groups Alt _m acting on the natural module of dimension d ≠ m − 1 or m − 2, and classical groups with natural module of dimension d over subfields of F _q .