  Let $p$ be an odd prime and $G$ be a nonabelian group of order $p^{n}$ with
the presentation $$<\alpha,\beta,\gamma\mid
\alpha^{p^{a}}=\beta^{p^{b}}=\gamma^{p^{c}}=1,
[\alpha,\gamma]=1,[\gamma,\beta]=\alpha^{p^{r}},[\alpha,\beta]=\gamma^{p^{e}}>,$$
where $n>a\geq b\geq c\geq 1$. Let $k$ be a field containing a primitive
$p^{a}$-th root of unity and $G$ act on the rational function field
$k(x_{h}:h\in G)$ by $g\cdot x_{h}=x_{gh}$ for all $g,h\in G$. In this note, we
prove that the fixed field $k(G)=k(x_{h}:h\in G)^{G}$ is rational over $k$. As
a corollary, we prove that if $k$ contains a primitive $p^{4}$-th root of unity
and $G$ is a nonabelian group of order $p^{5}$ generated by three elements,
then $k(G)$ is rational over $k$.
