  Let $X$ be a compact K\"ahler manifold. We study subgroups $G \le
\mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when
$\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We
show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies
$\ell_{\mathrm{vir}}(G) \le \dim X -\kappa(X)$, where $\kappa(X)$ is the
Kodaira dimension of $X$. Modulo the main conjecture of our previous work
concerning the essential nilpotency class, we obtain the same upper bound
$c_{\mathrm{vir}}(G) \le \dim X -\kappa(X)$ for the virtual nilpotency class
$c_{\mathrm{vir}}(G)$, together with a geometric description of the $G$-action
on $X$ when the equality holds.
