  Let $\rm{Aut}(p)$ denote the space of all self-fibre homotopy equivalences of
a principal $G$-bundle $p: E\rightarrow X$ of simply connected CW complexes
with $E$ finite. When $G$ is a compact connected topological group, we show
that there exists an inequality $$n-{\rm N}(p)\leq {\rm
Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_0)\leq n$$ for any space $X$, where $n$ is the
number of non-trivial rational homotopy groups of $G$ and ${\rm N}(p)$ is
defined in Section 2. In particular, ${\rm
Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_{0})=n$ if $p$ is a fibre-homotopy trivial
bundle and X is finite.
