  The mininal degree of a finite group G, mu(G), is defined to be the smallest
natural number n such that G embeds inside Sym(n). The group G is said to be
exceptional if there exists a normal subgroup N such that mu(G/N)>mu(G). We
will investigate the smallest exceptional p-groups, when p is an odd prime. In
1999 Lemiuex showed that there are no exceptional p-groups of order strictly
less than p^5 and imposed severe restrictions on the existence of exceptional
groups of order p^5. In fact he showed that if any were to exist, they must
come from central extensions of four isomorphism classes of groups of order
p^4. Then in 2007 he exhibited an example of an exceptional group of order p^5.
The author demonstrates the existence of two more exceptional groups arising in
such a fashion and rules out the possibility of the remaining case.
