Let a group $G$ act on a finite dimensional vector space $V$ over an
algebraically closed field $K$ of characteristic $p$. Then $\beta_{\sep}(G)$ is
the minimal number such that, for any $V$, the invariants of degree less or
equal than this number have the same separating properties as the whole
invariant ring $K[V]^{G}$. Derksen and Kemper have shown $\beta_{\sep}(G)\le
|G|$. We show $\beta_{\sep}(G)=|G|$ for $p$-groups and cyclic groups, and
$\beta_{\sep}(G)=\infty$ for infinite unipotent groups. We also show
$\beta_{\sep}(G)\le \beta_{\sep}(G/N)\beta_{\sep}(N)$ for a normal divisor $N$
of finite index.