On finite groups whose derived subgroup has bounded rank

Let $G$ be a finite group with derived subgroup of rank $r$. We prove that $\text{gzz}\le \mathrm{\mid }{G}^{\mathrm{\prime }}{\mathrm{\mid }}^{2r}$. Motivated by the results of I. M. Isaacs in \cite{isa} we show that if $G$ is capable then $\text{gz}\le \mathrm{\mid }{G}^{\mathrm{\prime }}{\mathrm{\mid }}^{4r}$. This answers a question of L. Pyber. We prove that if $G$ is a capable $p$-group then the rank of $G\mathrm{/}\mathbf{Z}(G)$ is bounded above in terms of the rank of $G^{\mathrm{\prime }}$.