  Let $G$ be a torsion-free, finitely-generated, nilpotent and metabelian
group. In this work we show that $G$ embeds into the group of orientation
preserving $C^{1+\alpha}$-diffeomorphisms of the compact interval, for all
$\alpha< 1/k$ where $k$ is the torsion-free rank of $G/A$ and $A$ is a maximal
abelian subgroup. We show that in many situations the corresponding $1/k$ is
critical in the sense that there is no embedding of $G$ with higher regularity.
A particularly nice family where this happens, is the family of
$(2n+1)$-dimensional Heisenberg groups, for which we can show that the critical
regularity equals $1+1/n$.
