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Suppose A is a 2-group with the following properties:

  1. $\lvert A \rvert = t^3$ with $t$ some even power of $2$;
  2. $A$ and $Z(A)$ (the center of $A$) are of exponent $4$;
  3. $\lvert Z(A) \rvert = t$ and $[A, A] = \sqrt{t}$;
  4. $[A, A]$ is the group of involutions in $Z(A)$;
  5. $A/[A, A]$ is elementary abelian.

I also have the following additional property: any two noncommuting involutions in $A$ are contained in a group isomorphic to $D_8$.

Can such groups be classified? Can one say much more about such groups (possibly under extra conditions)?

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    $\begingroup$ Surely the fact that $A$ has exponent $4$ implies that any two noncommuting involutions generate a group isomorphic to $D_8$? Or did you mean any two noncommuting elements of $A$ rather than involutions? I would expect there to be vast numbers of groups satisfying these conditions, and no hope of a classification. $\endgroup$
    – Derek Holt
    Commented Jan 6, 2021 at 13:07
  • $\begingroup$ @DerekHolt: you are right, I edited the question. I also have the following strong extra restrictions: there are $t + 1$ elementary abelian subgroups $A_0,\ldots, A_t$ of order $t$ such that for all different $i, j, k$, $A_iZ(A) \cap A_j = \{ 1 \}$ and $A_iA_j \cap A_k = \{ 1 \}$. Could this help ? $\endgroup$
    – THC
    Commented Jan 6, 2021 at 14:13
  • $\begingroup$ Does not this just amount to computing a certain (easily describable) subgroup in $H^2(\mathbb F_2^{5n};\mathbb F_2^n)$? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 6, 2021 at 18:30
  • $\begingroup$ @მამუკაჯიბლაძე: can you explain the details ? (Certainly, the extra conditions I mentioned to Derek Holt will be needed.) $\endgroup$
    – THC
    Commented Jan 6, 2021 at 22:22
  • $\begingroup$ I only mean that $A$ is a central extension of $A/[A,A]$ which is elementary abelian of order $2^{5n}$ by $[A,A]$ which is also elementary abelian of order $2^n$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 6, 2021 at 22:25

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