Group whose center is normality-large

Definition
A group whose center is normality-large is a group such that the intersection of its center with any nontrivial normal subgroup is nontrivial.

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Nilpotent group
 * Quasisimple group that is not simple (i.e., a perfect group with a nontrivial center, such that the quotient by the center is a simple group)

Weaker properties

 * Group in which every nontrivial normal subgroup contains a cyclic normal subgroup

Opposite properties

 * Centerless group