Does there exist a finitely presented, torsion-free group $G$ which has conjugacy classes of finite size greater than one?


This condition came up in a research project, and we would like to rule out the existence of such examples.


**Edit**: Sorry for the confusion. I meant to ask for an example of such $G$ such that *some* conjugacy class is of finite size greater than one. Colin Reid's comment shows that the Klein bottle group is an example.


Let me ask a follow-up: If $G$ is f.p. torsion-free and $G/Z(G)=\operatorname{Inn}(G)$ is torsion-free, then is every conjugacy class of $G$ either central or infinite?



Torsion-free groups with finite conjugacy classes
Such groups do not exist by Schur's theorem <https://chiasme.wordpress.com/2015/01/07/a-theorem-of-schur-on-commutator-subgroup/>


