Manifold with a nilpotent fundamental group.

Question

I am looking for a manifold with a nonabelian nilpotent fundamental group. I know the above terms, but I couldn't find out an example of that.

Answer 1

The Klein bottle has a fundamental group with a normal $\mathbb Z \oplus \mathbb Z$ subgroup and a $\mathbb Z / 2 \mathbb Z$ kernel.

Answer 2

Let $M$ be a compact homogeneous flat pseudo-Riemannian manifold. Then the fundamental group of $M$ is $2$-step nilpotent, and there are examples where it is non-abelian.

Answer 3

Probably an another example is here: The Heisenberg manifold, because the Heisenberg group is a nilpotent group.