Are there any nontrivial second Hurewicz homomorphisms for familiar compact 6-dimensional manifolds?

Question

Based on several computations I have done, it seems that the second Hurewicz homomorphism $$h:\pi_{2}(X)\rightarrow H_{2}(X;\mathbf{Z})$$ has a habit of being trivial. For instance, this seems to be the case for $X=S^{6}, \mathbf{RP}^{6},$ and $\mathbf{T}^{6}$, as well as for all compact symplectic groups $Sp(n)$, unitary groups $U(n)$, and special unitary groups $U(n)$, as well as the orthogonal groups $O(n)$ for $n\geq 4$ (which are the ones I care about for my problem).

This leaves me wondering: are there any compact, familiar, even-dimensional manifolds (dimension at least $6$) for which the second Hurewicz homomorphism isn't trivial?

If there aren't, would any of you happen to know any less familiar or less elementary examples that fit the bill?

Answer 1

Just take $X=S^2\times T^4$. To obtain such a manifold.

Answer 2

What about a simple connected 6 dimesional manifold whose 2nd homology group is not trivial, i.e $S^2\times S^4$.

By Kunneth formula we know that $H_2\neq 0$. And by Hurewicz theorem, $h$ is an isomorphism.