Group having an abelian normal non-central subgroup

Definition
A group having an Abelian normal non-central subgroup is a group satisfying the following equivalent conditions:


 * 1) There exists an Abelian normal subgroup not contained inside the center
 * 2) There exists an Abelian normal subgroup properly containing the center

Stronger properties

 * non-Abelian nilpotent group
 * non-Abelian supersolvable group
 * non-Abelian group in which maximal among Abelian normal implies self-centralizing