Here are some additional details for the answer of M. Farrokhi. In the paper " On a conjecture of Hanna Neumann", B.H.Neumann constructs a certain group $G$. There are generators $a_1,\dotsc,a_4$, and additional elements defined in terms of these as follows:
\begin{align*}
b_{12} &= [a_1,a_2] = [a_2,a_1] \\
b_{13} &= [a_1,a_3] = [a_3,a_1] \\
b_{14} &= [a_1,a_4] = [a_4,a_1] \\
b_{23} &= [a_2,a_3] = [a_3,a_2] \\
b_{24} &= [a_2,a_4] = [a_4,a_2] \\
b_{34} &= [a_3,a_4] = [a_4,a_3] \\
c_1 &= [a_2,b_{34}] = [a_4,b_{23}] \\
c_2 &= [a_3,b_{14}] = [a_4,b_{13}] \\
c_3 &= [a_4,b_{12}] = [a_1,b_{24}] \\
d &= [a_1,[a_2,[a_3,a_4]] = [a_2,[a_3,[a_4,a_1]] = [a_3,[a_4,[a_1,a_2]]
\end{align*}

There are some relations implicit in the above equations. There are also additional relations as follows:

- $a_i^2=b_{jk}^2=c_l^2=d^2=1$
- All commutators $[a_i,b_{jk}]$ that have not already been listed, are trivial.
- $[a_i,c_j]=1$ whenever $i\neq j$, and $[a_i,d]=1$

We can define a map $\phi\colon\{0,1\}^{14}\to G$ by
$$ \phi(u) = a_1^{u_1}\dotsb a_4^{u_4} b_{12}^{u_5} \dotsb b_{34}^{u_{10}}
c_1^{u_{11}}c_2^{u_{12}}c_3^{u_{13}}d^{u_{14}}
$$
One can check that this is bijective, and one can write formulae for the permutations of $\{0,1\}^{14}$ corresponding to right multiplication by the elements $a_i$, $b_{jk}$, $c_l$ and $d$. In particular, this proves that $|G|=2^{14}$. One can also check that
$$ [b_{12},b_{34}] = [b_{13},b_{24}] = [b_{14},b_{23}] = d \neq 1, $$
so $G$ is not metabelian.

In the paper "On certain varieties of groups", Macdonald states that it is easy to verify that the above group has $[x,y]^2=1$ for all $x,y\in G$. I don't see how to prove this myself. However, I have checked it by computer for 10000 randomly chosen pairs $(x,y)$, so it must be true.