  Let $p$ be a prime number, $\Bbbk$ a field of characteristic $p$ and $G$ a
finite $p$-group. Let $V$ be a finite-dimensional linear representation of $G$
over $\Bbbk$. Write $S = \mathrm{Sym} V^*$. For a class of $p$-groups which we
call generalised Nakajima groups, we prove the following: \begin{enumerate}
\item The Hilbert ideal is a complete intersection. As a consequence, for the
case of generalised Nakajima groups, we prove a conjecture of Shank and Wehlau
(reformulated by Broer) that asserts that if the invariant subring $S^G$ is a
direct summand of $S$ as $S^G$-modules then $S^G$ is a polynomial ring. \item
The Hilbert ideal has a generating set with elements of degree at most $|G |$.
This bound is conjectured by Derksen and Kemper. \end{enumerate}
