Is there a presentation with two generators and two relators for the group $C\_4 \cdot D\_8$?
This group is of order 32 and its IdSmallGroup in GAP is [32,15].
Also it has the following presentation with 2 generators and 3 relators:


$\langle x,y \;|\; y^x=y^3, (y^2)^x=y^{-2}, yx^{-1}y=x^3\rangle$



A possible presentation with 2 generators and 2 relators for $C_4 \cdot D_8$
$$\langle x,y \mid y^x=y^3, yx^{-1}y=x^{-5}\rangle.$$


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. 


