Nilpotent group with rationally powered abelianization need not be rationally powered

Statement
It is possible to have a nilpotent group $$G$$ such that the abelianization of $$G$$ is a rationally powered group (which in this case means it is a vector space over $$\mathbb{Q}$$) but such that $$G$$ is not rationally powered over any prime. In fact, we can select $$G$$ to be such that it has $$p$$-torsion for every prime $$p$$.

This also shows that the derived subgroup of a nilpotent group need not be a quotient-powering-faithful subgroup.

Opposite facts

 * The group is forced to be divisible for all primes that the abelianization is divisible by. See Nilpotent group is divisible by a prime iff its abelianization is and iff all lower central series quotients are.

Proof
Let $$G$$ be the quotient group of $$UT(3,\mathbb{Q})$$ (the unitriangular matrix group of degree three over the field of rational numbers) by a subgroup $$\mathbb{Z}$$ inside its center (which is a copy of $$\mathbb{Q}$$). Explicitly, we can think of $$G$$ as matrices of the form:

$$\{ \begin{pmatrix} 1 & a_{12} & \overline{a_{13}} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{23} \in \mathbb{Q}, \overline{a_{13}} \in \mathbb{Q}/\mathbb{Z} \}$$

with the matrix multiplication defined as:

$$\begin{pmatrix} 1 & a_{12} & \overline{a_{13}} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix}\begin{pmatrix} 1 & b_{12} & \overline{b_{13}} \\ 0 & 1 & b_{23} \\ 0 & 0 & 1 \\\end{pmatrix} = \begin{pmatrix} 1 & a_{12} + b_{12} & \overline{a_{12}b_{23}} + \overline{a_{13}} + \overline{b_{13}} \\ 0 & 1 & a_{23} + b_{23} \\ 0 & 0 & 1 \\\end{pmatrix}$$

where $$\overline{a_{12}b_{23}}$$ is understood to be the image of $$a_{12}b_{23}$$ under the quotient map $$\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$$.

The center of $$G$$ coincides with its derived subgroup, and is:

$$\left \{ \begin{pmatrix} 1 & 0 & \overline{a_{13}} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix} \mid \overline{a_{13}} \in \mathbb{Q}/\mathbb{Z} \right \}$$

The inner automorphism group and the abelianization are therefore both isomorphic to $$\mathbb{Q} \times \mathbb{Q}$$, which is rationally powered. However, the group as a whole has $$p$$-torsion for all primes $$p$$.