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Here's a one-relator ("large") example of a torsion free group such that the group of inner automorphisms has torsion:

$$\langle a_{1},a_{2},a_{3} : a_{1}^{2}=a_{2}^{3}, a_{2}^{5}=a_{3}^{7}\rangle$$.

This group is one-relator on $a_{1}$ and $a_{3}$, and by a result of Murasugi is torsion-free. The center of the group is generated by $a_{1}^{10}$.

This fact is mentioned in the first paragraph of this readily googlable paper of James McCool: A class of one-relator groups with centre, Bulletin of the Australian Mathematical Society, Volume 44, Issue 2, 2009.

The result is proved in this other readily googlable paper of Medskin, Pietrowski and Steinberg: One relator groups with center, Journal of the Australian Mathematical Society, Volume 16, 1973.

The result of Murasugi can be found in: The center of a group with a single defining relation, Math. Ann., 155, 1964.

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Here's a one-relator example:

$$\langle a_{1},a_{2},a_{3} : a_{1}^{2}=a_{2}^{3}, a_{2}^{5}=a_{3}^{7}\rangle$$.

This group is one-relator on $a_{1}$ and $a_{3}$, and by a result of Murasugi is torsion-free. The center of the group is generated by $a_{1}^{10}$.

This fact is mentioned in the first paragraph of this readily googlable paper of James McCool: A class of one-relator groups with centre, and Bulletin of the Australian Mathematical Society, Volume 44, Issue 2, 2009.

The result is proved in this other readily googlable paper of Medskin, Pietrowski and Steinberg: One relator groups with center, Journal of the Australian Mathematical Society, Volume 16, 1973.

The result of Murasugi can be found in: The center of a group with a single defining relation, Math. Ann., 155, 1964.

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