Smallest nonabelian quotients of surface braid groups

We give a sharp lower bound on the size of nonabelian quotients of the surface braid group $B_n({\mathrm{\Sigma }}_g)$ and classify all quotients that attain the lower bound: Depending on $n$ and $g$, a quotient of minimum order is either a symmetric group or a 2-step nilpotent $p$-group.