How to find the discrete Heisenberg group in this group given by a small presentation?

Question

Consider the group $G$ given by the following presentation:

$$G=\langle x,y\mid x^{-1}y^2xy^2=x^{-2}yx^{-2}y^3=1\rangle.$$

In this slides it is noted that this is a torsion-free polycyclic group, which is virtually the discrete Heisenberg group, see p.16.

I am interested in the last piece of information. According to the statement, one should be able to find a copy of the Heisenberg group $H$ in $G$, such that $G/H$ is finite.

One possible presentation of the Heisenberg group is given by $$H=\langle a,b\mid [a,[a,b]]=[b,[a,b]]=1\rangle.$$

How to find such a copy of $H$ in $G$ with finite quotient? Can this be done by hand or are the computer-base algorithms that find such copies? I am not an expert in group theory, so maybe this is obvious?

Answer 1

In the slides you link, it is mentioned this example comes from Lindsay Jennae Soelberg's Master's thesis. She mentions "[using] Magma as a tool to show that $G_Z$ is torsion-free [following] an idea of Derek Holt, given in response to a question at MathOverflow", which I suspect may refer to this answer.

To quote the thesis (p24):

Using Magma we search for a subgroup of $G_Z$ of small index and with an especially easy rewrite system. It yielded the following subgroup which met the qualifications we required. Let $H$ be the subgroup generated by the three elements $h_1 := yxy^{−1}x^{−1}$, $h_2 := y^2$, and $h_3 := x^4$. The group $H$ has index $8$ in $G_Z$, it is a normal subgroup, and it is given by the finite presentation $$H = \langle h_1, h_2, h_3 : h_2 \textrm{ is central, and } h_3h_1 = h_1h_3h_2^8\rangle.$$

From this presentation, it becomes clear that $K := \langle h_1,h_3\rangle$ is isomorphic to the Heisenberg group, has index $8$ in $H$ and index $64$ in $G_Z$. Do note, however, that while $K$ is normal in $H$, it is not normal in $G$.