  We find a new exact $\Lambda$-vacuum solution in pure Gauss-Bonnet gravity
with NUT charge in six dimension with horizon having product topology $S^{(2)}
\times S^{(2)}$. We also discuss its horizon and singularity structure, and
consequently arrive at a parameter window for its physical viability. It should
be noted that all NUT black hole solutions in higher dimensions have product,
instead of spherical, topology. We prove, in general, that it is because of the
radial symmetry of the NUT spacetime; i.e. in higher dimensions NUT spacetime
cannot maintain radial symmetry unless horizon has $S^{(2)}$ or its product
topology. On the way we also prove a general result for spherical symmetry that
when null energy condition is satisfied, one has then only to solve a first
order equation to get a vacuum or $\Lambda$-vacuum solution.
