On a compact five-manifold, the Stiefel-Whitney number $w_2w_3$ can be nonzero. An example is the manifold $SU(3)/SO(3)$, and also another example is a $\mathbb{CP}^2$ bundle over a circle where the holonomy is given by the complex conjugation automorphism of $\mathbb{CP}^2$.

My question is whether the Stiefel-Whitney number $w_1^2w_3$ can be nonzero on a compact five-manifold. I haven't found either a simple proof that it is zero, or a simple example where it is nonzero.