  We prove that the symmetric group $S_n$ is the smallest non-cyclic quotient
of the braid group $B_n$ for $n=5,6$ and that the alternating group $A_n$ is
the smallest non-trivial quotient of the commutator subgroup $B_n'$ for $n =
5,6,7,8$. We also give an improved lower bound on the order of any non-cyclic
quotient of $B_n$.
