Is there an infinite finitely generated (non-cyclic) $p$-group $G$ such that the intersection of all nontrivial subgroups of $G$ is a nontrivial subgroup?



Existence of an infinite finitely generated $p$-group with nontrivial intersection of nontrivial subgroups
An example is given [here](https://mathoverflow.net/questions/102208/finitely-generated-infinite-p-group). And here are some more words to make MO happy.


