Let p be an odd prime and k an arbitrary field of characteristic not p. 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.
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