Maximal linear groups induced on the Frattini quotient of a p-group

Abstract

Let p>3 be a prime. For each maximal subgroup H⩽GL(d,p) with |H|⩾p3d+1, we construct a d-generator finite p-group G with the property that Aut(G) induces H on the Frattini quotient G/Φ(G) and |G|⩽pd42. A significant feature of this construction is that |G| is very small compared to |H|, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on G/Φ(G), the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.