  Let p be an odd prime. The lattice of all normal subgroups and the terms of
the lower and upper central series are determined for all metabelian p-groups
with generator rank d=2 having abelianization of type (p,p) and minimal defect
of commutativity k=0. It is shown that many of these groups are realized as
Galois groups of second Hilbert p-class fields of an extensive set of quadratic
fields which are characterized by principalization types of p-classes.
