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Is there a finitely generated group $G$, with a proper subgroup $H$, such that $$G=\bigcup_{g\in G}gHg^{-1}$$

I know that this is not the case for a finite $G$: Union of the conjugates of a proper subgroup

I also found an example for an infinite $G$: https://mathoverflow.net/questions/34044/group-cannot-be-the-union-of-conjugates

but (as far as I know) the example given, $GL_2(\mathbb{C})$ is not finitely generated.

I've also seen that there are finitely generated groups with proper subgroups of infinite index (that are not abelian): give an example of finitely generated not abelian group which has subgroup of infinite index and not finitely generated.

but the groups given are above my level of understanding, and thus I'm not sure if it helps me with my question.

And for abelian groups this obviously fails as for all $g\in G$ we get $gHg^-1=H$

Unfortunately even with all of that, I wasn't able to figure out myself the answer to the question. Any help would be appreciated.

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    The answer to the question is yes. In the answer to the mathoverflow question, Primoz mentions $2$-generated examples, arising from a paper of Ivanov and Ol'shanskii. But i you are looking for "easy "examples, then it might be more difficult! – Derek Holt Nov 10 '15 at 12:28
  • @Derek Holt Oh I see. That's interesting. Though guess I'll need to come back to it aftrr I've studied the subject for a bit longer.. – Nescio Nov 10 '15 at 12:34

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