  We consider for $d\geq 1$ the graded commutative $\mathbb{Q}$-algebra
$\mathcal{A}(d):=H^*(\operatorname{Hilb}^d(\mathbb{C}^2);\mathbb{Q})$, which is
also connected to the study of generalised Hurwitz spaces by work of the first
author. These Hurwitz spaces are in turn related to the moduli spaces of
Riemann surfaces with boundary. We determine two distinct, minimal sets of
$\lfloor d/2\rfloor$ multiplicative generators of $\mathcal{A}(d)$.
Additionally, we prove when the lowest degree generating relations occur. For
small values of $d$ we also determine a minimal set of generating relations,
which leads to several conjectures about the necessary generating relations for
$\mathcal{A}(d)$.
