$$\langle x,y \mid y^x=y^3, yx^{-1}y=x^{-5}\rangle.$$

I found that presentation by trial and error, but minimal presentations are known for $2$-groups of order up to $64$. See

Sag, T. W.; Wamsley, J. W. Minimal presentations for groups of order $2n$, $n \le 6$, J. Austral. Math. Soc. 15 (1973), 461–469.

$$\langle x,y \mid y^x=y^3, yx^{-1}y=x^{-5}\rangle.$$

I found that presentation by trial and error, but minimal presentations are known for $2$-groups of order up to $64$. See

Sag, T. W.; Wamsley, J. W. Minimal presentations for groups of order $2^n$, $n \le 6$, J. Austral. Math. Soc. 15 (1973), 461–469.

$$\langle x,y \mid y^x=y^3, yx^{-1}y=x^{-5}\rangle.$$

$$\langle x,y \mid y^x=y^3, yx^{-1}y=x^{-5}\rangle.$$

I found that presentation by trial and error, but minimal presentations are known for $2$-groups of order up to $64$. See

Sag, T. W.; Wamsley, J. W. Minimal presentations for groups of order $2n$, $n \le 6$, J. Austral. Math. Soc. 15 (1973), 461–469.