For each prime p and positive integer n there are exactly two (up to isomorphism) extraspecial groups of order p1+2n.
Then the Arf invariant of this quadratic form can be used to distinguish the two extraspecial groups.
This is a special case of a classification of p-groups with cyclic centers and simple derived subgroups given in (Newman 1960).
The classification of countably infinite extraspecial groups is very similar to the finite case, (Newman 1960), but for larger cardinalities even basic properties of the groups depend on delicate issues of set theory, some of which are exposed in (Shelah & Steprāns 1987).
The nilpotent groups whose center is cyclic and derived subgroup has order p and whose conjugacy classes are at most countably infinite are classified in (Newman 1960).