  We enumerate central and medial quasigroups of order less than $128$ up to
isomorphism, with the exception of those quasigroups that are isotopic to
$C_4\times C_2^4$, $C_2^6$, $C_3^4$ or $C_5^3$. We give an explicit formula for
the number of quasigroups that are affine over a finite cyclic group.
