Exceptional p-groups of order p^5

Abstract

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.