  We give refined estimates for the discrete time and continuous time versions
of some basic random walks on the symmetric and alternating groups $S_n$ and
$A_n$. We consider the following models: random transposition, transpose top
with random, random insertion, and walks generated by the uniform measure on a
conjugacy class. In the case of random walks on $S_n$ and $A_n$ generated by
the uniform measure on a conjugacy class, we show that in continuous time the
$\ell^2$-cuttoff has a lower bound of $(n/2)\log n$. This result, along with
the results of M\"uller, Schlage-Puchta and Roichman, demonstrates that the
continuous time version of these walks may take much longer to reach
stationarity than its discrete time counterpart.
