# A question on UCS p-groups(2)

This is a follow up to this question

Let $$G_{1}$$ and $$G_{2}$$ be two finite UCS p-groups with the following conditions:

1- $$\vert G_{1}\vert=\vert G_{2}\vert=p^{2n}$$;

2- $$\Phi(G_{1})\cong\Phi(G_{2})\cong\underbrace{\mathbb{Z}_{p}\times\mathbb{Z}_{p}\times\dots\times\mathbb{Z}_{p}}_{n\,\,times}$$;

3- $$G_{1}$$ and $$G_{2}$$ have the same number of minimal subgroups.

Can we prove that $$G_{1}\cong G_{2}$$? Is there any counterexamples?

• I added some conditions to the problem. However if somebody has any result on the first question, I am eagerly waiting to see it, since my main question is the first one. – H.Shahsavari Feb 1 at 18:22
• Counterexamples for $p=2$ and $n=2k+1\ge5$ are provided in: Graham Higman, Suzuki $2$-groups, Illinois J. Math. 7 (1963), 79--96 – Richard Lyons Feb 3 at 21:40
• @ Professor Lyons. Many thanks for your valuable comment. – H.Shahsavari Feb 8 at 17:38