Extraspecial groups p±1+2n
An extraspecial group is a p-group G with center Z of order p such that G/Z is (non-trivial and)
elementary abelian. There are exactly two such groups for every prime power of the form
For odd p, the '+' one has exponent p, and the '-' one exponent p2.
When n=1, p+1+2=Hep is the Heisenberg
group, and p-1+2=Hep=Cp2⋊Cp
the other non-abelian group of order p3.