/*
HS:2 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^5,
(x,y)^3,
(x,y^2)^4,
((x*y)^4*x*y^-2*x*y^-1*x*y^-1*x*y^-2)^2,
(x,y*x*y*x*y^2*x*y^-2*(x*y^2)^2*x*y^-1),
(x,y*x*y*x*y^-2*x*y^-1*(x*y^-2)^3),
(x,y^-1*x*y^-1*x*y^2*x*y*(x*y^2)^3)
>;
c:=x;d:=y;

H:=sub<G|x,y*(x,y),(x,y^2)^2,(x,y^-2)^2>;
// H = C_G(x), of index 1100.
K:=sub<G|x,x^(y^2)>;
// K = D8, of index 11088000.
L:=sub<G|x,y*x*y*x*y^-1>;
// L = M22:2, of index 100.

/*
Proofs:
Proof 1:
Relations imply that G/G' = 2 and G is generated by the conjugates of x.
Relations also show that H centralises x.
Coset enumeration shows that H has index 1100 in G, the same as what we
expect in the known image HS:2.
Thus G is a proper cover of HS:2, namely HS:2, 2"HS:2 or 2"HS"2.
We rule out 2"HS"2 as x lifts to an element of order 4, and we have x^2=1.
We rule out 2"HS:2 as x lifts to an involution whose centraliser has index
2200, whereas C_G(x) has index 1100.

Proof 2:
Coset enumeration over <x,x^(y^2)> = D8.
Requires considerably more computing power.
The index we obtain is 11088000.

The following should work.
Index(G,H:Print:=2,Hard:=true,CosetLimit:=10^6,Grain:=10^6);
Index(G,K:Print:=2,Hard:=true,CosetLimit:=33*10^6,Grain:=10^6);
Index(G,L:Print:=2,Hard:=true,CosetLimit:=10^6,Grain:=10^6);
*/