On exisitance a finite group

Question

Do there exists group $G$ such that

$$\dfrac{G}{Z(G)}\cong\langle a,b,x\mid a^8=b^8=x^4=1,a^4=b^4=x^2, [a,x]=[b,x]=1, [a,b,b]=[a,b,a]=[b,a,a]=[b,a,b]=1\rangle.$$

Answer 1

No, there is not.

As the (gap) tag was given this is presumably intended as a request on how to do this calculation. First construct the group and convert it to a PcGroup:

f:=FreeGroup("a","b","x");
rels:=ParseRelators(f,"a8=b8=x4=1,a4=b4=x2,[a,x]=[b,x]=1,[[a,b],b]=[[a,b],a]=[[b,a],a]=[[b,a],b]=1");
g:=f/rels;
Size(g);
g:=Image(IsomorphismPcGroup(g));

The command IsCentralFactor tests whether a group is capable, in this case it returns false, thus the group is not capable.