$P_2$-groups are not always $P_3$-groups : a group by Blackburn

Question

I am trying to find the group refered to after Collary 8 in this paper ¹. The relevant quote is:

A group constructed by Blackburn [3; III.10.15], for which $|G| = p^{p+1}$, $ |\Omega_1 (G)| = p^{p-1}$ and $|\mho_1(G)| = p$ shows that $P_2$-groups are not always $P_3$-groups.

3. B. HUPPERT, “Endliche Gruppen I,” Springer-Verlag, Berlin, 1967.

I do not have access to the the text cited (translated as "Finite Groups"). Does anyone know the group Mann is refering to?

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For context $\mho_n(G) := \langle g^{p^n} \mid g \in G \rangle$ and $\Omega_n(G) := \langle g \mid g^{p^n} = 1 \rangle$.

$G$ is called a $P_2$-group if $ \Omega_n(S) = \{ g \mid g^{p^n} = 1 \}$ for all $n$ and every section $S$ of $G$.

And $G$ is called a $P_3$-group if $| \mho_n(S) | =[S:\Omega_n(S)] $ for all $n$ and every section $S$ of $G$,

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Reference:

(1) Mann, A. (1976). The power structure of p-groups. I. Journal of Algebra, 42(1), p.125.

Answer 1

My German is rusty, but I am instead attaching the two relevant pages of the book of Huppert. It's a monstrous book. It appears from a cursory glance that all the notation is pretty ordinary, so I hope it is understandable.

It's in German, which I unfortunately can't help you with.