@YCor: No. I am not trying to compute the cohomology. I am interested in mathoverflow.net/questions/209438/… It is known that the number $R_G$ defined in the latter link is the minimum number of relations needed to present the finite $p$-group $G$ as a (topological) quotient of the free pro-$p$-group of rank $d_G$ the minimum number of generators of $G$.

@YCor: Is it possible to present the free pro-p-group as a free group quotient "without referring to the free pro-p-group itself"? Or maybe better to say, find a subset $X$ in $R_G$ which is minimal with respect to the inclusion such that $\langle \langle X \rangle \rangle=\langle \langle R_G \rangle \rangle$. Anyway, this may have a more or less trivial answer as the problem is to how define "explicit".

@YCor: Can you write down the presentation for $\mathbb{Z}_p$ in your way?

@ahulpke: Thanks for the information given by email! I will be happy if you could enter them as a comment!

@DerekHolt: Thanks for the information.

@ahulpke Thanks. Could you please let me know your opinion about the following question: mathoverflow.net/questions/209438/…

@DerekHolt Thanks. Could you please let me know your opinion about the following question: mathoverflow.net/questions/209438/…

@Steve D: As you gave the above comment, I would like to know if you have any comment on this question: mathoverflow.net/questions/209438/…

In the reference, $2n$ should be $2^n$.

@ Many thanks. Could you please give a bit explanation how you get this presentation?