Isolated singularities with large class group

Let $d\geq3$ and $g\geq1$ be integers. Using a geometric construction involving the symmetric product of a projective curve, we exhibit a $d$-dimensional complete local normal domain over $\mathbb{C}$ with an isolated singularity such that its class group contains a copy of $(\mathbb{R}/\mathbb{Z})^{\oplus 2g}$ as subgroup.