  In this article we study graded Clifford algebras with a gradation preserving
action of automorphisms given by $H_p$, the Heisenberg group of order $p^3$
with $p$ prime. After reviewing results in dimensions 3 and 4, we will
determine the graded Clifford algebras that are AS-regular algebras of global
dimension 5 and generalize certain results to arbitrary dimension $p$. More
specifically, we prove that there is a nice correspondence between
$\mathbb{P}^1_{\mathbb{F}_p}$ and the algebras isomorphic to the quantum space
$\mathbb{C}_{-1}[x_0,\ldots,x_{p-1}]$.
