How do I calculate the co-rank of a group $$G=\langle a,b,c,\dots,z\mid a^2b^2c^2\dots z^2=1\rangle,$$ that is, finitely generated group with one relation?
By co-rank, I mean the maximum rank of a free homomorphic image of the group.
(Motivation: This is the fundamental group of a non-oriented surface. For orientable surface $M^2_g$ this value is $g$, but I need to know it for non-orientable surface $M^2_h$.)
In case it helps, I know that abelianization ($G$ made abelian) of this group is $Z^{n-1}\oplus(Z/2Z)$, where $n$ is the number of generators.