Fixed-class capable group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Suppose $c$ is a positive number. A group $G$ is a class $c$-capable group if there exists a group $K$ such that $G\cong K/Z_{c}(K)$ where $Z_{c}(K)$ denotes the $c^{th}$ member of the upper central series of $K$.

Note that $Z_{1}(K)=K$ is the center and class 1-capable group is the same notion as capable group.