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 p1+2n, denoted p+1+2n and p-1+2n. 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.
D4=2+ 1+2Q8=2- 1+2He3=3+ 1+23- 1+22+ 1+42- 1+4He5=5+ 1+25- 1+22+ 1+62- 1+63+ 1+43- 1+4He7=7+ 1+27- 1+2