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.Comment: 23 page
