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Although there are p^{2n^3/27} groups G of order p^n
(by work of Higman and Sims), and Aut(G) can have arbitrary
quotients (by work of Heineken and Liebeck), most p-groups
G have very small automorphism groups (by Helleloid and Martin).
This talk considers p-groups G where G, Phi(G) and 1
are the only characteristic subgroups. These groups seem to have
large and interesting automorphism groups seemingly
related to maximal irreducible linear groups. Their study raises
questions about the representation theory of exterior squares. This
is joint work with Csaba Schneider and Peter Pálfy.
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