Finite 2-groups with a nonabelian Frattini subgroup of order 16

According to a classical result of Burnside, if G is a finite 2-group, then the Frattini subgroup Φ(G) of G cannot be a nonabelian group of order 8. Here we study the next possible case, where G is a finite 2-group and Φ(G) is nonabelian of order 16. We show that in that case Φ(G) ≅ M × C₂, where M ≅ D₈ or M ≅ Q₈ and we shall classify all such groups G (Theorem A). Received: 16 February 2005; revised: 7 March 2005