In the answer to MO question 132247, it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.

Q.What are the quotient groups of the lower central series of the genus $g$ surface group $\Pi_g$, namely $$\Pi_g = \langle a_1, \ldots, a_g, \, b_1, \ldots, b_g \; | \; [a_1, \, b_1] \cdots [a_g, \, b_g]=1 \rangle?$$

I guess that it is a very classical subject with a huge literature, but I am not an expert in the area, so any pointer to some relevant book/article will be higly appreciated.