Suppose one has a surface $S$ of genus $g\ge 2$, $x\in S$, and $G=\pi(S,x)$ the fundamental group. There is a well-known description of the group with generators and relations as $$G=\langle a_1,b_1,\dots, a_g,b_g \ | [a_1,b_1]\cdots[a_g,b_g]=1\rangle$$
Now, it is know that the usual homology $H_2(S,\mathbb{Z})\cong \mathbb{Z}$ and that $$H_2(G,\mathbb{Z})\cong H_2(S,\mathbb{Z})$$ using group cohomology.
On the other hand the group cohomology can be descrived using the so called bar resolution, whose elements $z$ are in $\mathbb{Z}[G]\otimes \mathbb{Z}[G]$ such that $\delta_2(z)=0$, where $\delta_2(g\otimes h)= g+h-gh$, where $\delta_2$ goes to $\mathbb{Z}[G]$ (and then quotienting by the image of a $\delta_3$ analogous to $\delta_2$).
My question is: what is the generator of $H_2(G,\mathbb{Z})$ expressed as an element in $\mathbb{Z}[G]\otimes \mathbb{Z}[G]$ with the generators $a_i$, $b_i$, $i=1,\dots,g$ given above?
I am studying group cohomology using K.S. Brown book "Cohomology of groups".