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One knows that any countable group can be embedded in a 2-generated group. Also, any finite group is a subgroup of a 2-generated finite group, namely by the Cayley embedding. Does the same hold if we restrict to finite $p$-groups, where $p$ is some prime, i.e. is any finite $p$-group a subgroup of a 2-generated finite $p$-group?

Note that the minimal number of generators of the Sylow subgroup of the symmetric groups grow with their order. Thus the Cayley embedding would not suffice.

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This is proved in

Neumann, B. H. and Neumann, H. (1959), Embedding Theorems for Groups. Journal of the London Mathematical Society, s1-34: 465–479. doi:10.1112/jlms/s1-34.4.465

At least, it says in the introduction that they prove this. For some reason I don't seem to be able to access the full article online at the moment.

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