  Let $p$ be an odd prime, and let $k$ be an arbitrary field of characteristic
not $p$. In this article we determine the obstructions for the realizability as
Galois groups over $k$ of all groups of orders $p^5$ and $p^6$, that have an
abelian quotient obtained by factoring out central subgroups of order $p$ or
$p^2$. These obstructions are decomposed as products of $p$-cyclic algebras,
provided that $k$ contains certain roots of unity.
