On exceptional groups of order p^5
  A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient.   The smallest examples of exceptional p-groups have order p^5. For an odd prime p, we classify all pairs (G,Q) where G has order p^5 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 p^5 with at least one exceptional quotient tends to 1/2. 
John R. Britnell, Neil Saunders and Tony Skyner
math.GR