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Two elements in a group $G$ are said to $z$-equivalent or to be in the same$z$-class if their centralizers are conjugate in $G$. In \cite{kkj}, it wasproved that a non-abelian $p$-group $G$ can have at most $\frac{p^k-1}{p-1} +1$number of $z$-classes, where $|G/Z(G)|=p^k$. In this note, we characterize the$p$-groups of conjugate type $(n,1)$ attaining this maximal number. As acorollary, we characterize $p$-groups having prime order commutator subgroupand maximal number of $z$-classes.

z-Classes in finite groups of conjugate type (n,1)