  For finite p-groups P of class 2 and exponent p the following are invariants
of fully refined central decompositions of P: the number of members in the
decomposition, the multiset of orders of the members, and the multiset of
orders of their centers. Unlike for direct product decompositions, Aut P is not
always transitive on the set of fully refined central decompositions, and the
number of orbits can in fact be any positive integer. The proofs use the
standard semi-simple and radical structure of Jordan algebras. These algebras
also produce useful criteria for a p-group to be centrally indecomposable.
