Finite group schemes of $p$-rank $\leq1$

Let $\mathcal{G}$ be a finite group scheme over an algebraically closed field $k$ of characteristic ${\rm char}(k)=p\geq 3$. In generalization of the familiar notion from the modular representation theory of finite groups, we define the $p$-rank $\mathsf{rk}_p(\mathcal{G})$ of $\mathcal{G}$ and determine the structure of those group schemes of $p$-rank $1$, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height $1$, which correspond to restricted Lie algebras. Our results show that group schemes of $p$-rank $\leq 1$ are closely related to those being of finite or domestic representation type.