On exceptional groups of order p5

The smallest examples of exceptional p-groups have order p5. For an odd prime p, we classify all pairs (G,Q) where G has order p5 and Q is a distinguished quotient. (The case p=2 has already been treated by Easdown and Praeger.) We establish the striking asymptotic result that as p increases, the proportion of groups of order p5 with at least one exceptional quotient tends to 1/2.