According to the classification of finite non-abelian groups of order $p^4$ in Burnside's book "The theory of finite groups, 1897, pages 87-88" one of the types of these groups is the following family: $$G(\alpha):=\langle a, b, c \mid a^{p^2}=b^p=c^p=1, a^b=a^{p+1}, a^c=ab, b^c=a^{\alpha p}b\rangle,$$ where $p>3$ and $\alpha=0$, $\alpha=1$ or $\alpha$=any non-residue (mod $p$).
What does "$\alpha$=any non-residue (mod $p$)" mean?
Thanks for your help!
This is short for quadratic non-residue $\mod p$, meaning that $x^2 \equiv \alpha \mod p$ has no solution.