on the groups satisfying the converse of schur's theorem

a famous theorem of schur states that for a group $g$ finiteness of $g/z(g)$‎ ‎implies the finiteness of $g'.$ the converse of schur's theorem is an interesting problem which has been considered by some‎ ‎authors‎. ‎recently‎, ‎podoski and szegedy proved the truth of the converse of schur's theorem for capable groups‎. ‎they also established an explicit bound for the index of the center of such groups‎. ‎this paper is devoted to determine some families of groups among non-capable groups which satisfy the converse of schur's theorem and at the same time admit the podoski and szegedy's bound as the upper bound for the index of their centers‎.