Fixed-class capable group

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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.