Let $G$ be a group of order $p$ acting linearly on $A=\mathbb{Z}/p\mathbb{Z}[x_1,\ldots,x_r]$. Whether there exists a formula for the Hilbert-Samuel multiplicity of the unique homogeneous maximal ideal of $A^G$?

Let $G$ be a group of order $p$ acting linearly on $A=\mathbb{Z}/p\mathbb{Z}[x_1,\ldots,x_r]$. Does there exist a formula for the Hilbert-Samuel multiplicity of the unique homogeneous maximal ideal of $A^G$?