As far as I know, the Erlangen program, strictly speaking, describes connected (See page 138 of R.W.Sharpe) homogeneus manifolds $X$ as $G/H$ where $G$ is a Lie group considered as the "automorphism group" of a geometry on $X$ and $H$ is the stabilizer of a point.

First of all, the space $\mathbb{Z}^d$ is a non-connected zero dimensional manifold. Thus the "full symmetry group" as a manifold (even as a Riemannian manifold, being zero dimensional) is simply the full symmetric group (i.e. set theoretic permutations) $\mathfrak{S}(\mathbb{Z}^d)$, and $H$ the stabilizer of any point.

But you didn't say which structure on $\mathbb{Z}^d$ you want the symmetries to preserve...

If you want to preserve the distance induced by the Euclidean norm on $\mathbb{R}^d$, then you can take $G=O(d,\mathbb{Z})\ltimes\mathbb{Z}^d$ and $H=O(d,\mathbb{Z})$. [I see that S.Carnahan has posted the same suggestion right before me. Edit: and also Will Sawin]

As far as I read (See page 138 of R.W.Sharpe), the Erlangen program, strictly speaking, describes connected homogeneus manifolds $X$ as $G/H$ where $G$ is a Lie group considered as the "automorphism group" of a geometry on $X$ and $H$ is the stabilizer of a point.

First of all, the space $\mathbb{Z}^d$ is a non-connected zero dimensional manifold, so I don't know how much we can say it fits the Erlangen program. Anyways, the "full symmetry group" as a manifold (even as a Riemannian manifold, being zero dimensional) is then simply the full symmetric group (i.e. set theoretic permutations) $G=\mathfrak{S}(\mathbb{Z}^d)$, and $H$ the stabilizer of any point.

But you didn't say which structure on $\mathbb{Z}^d$ you want the symmetries to preserve...

If you want to preserve the distance induced by the Euclidean norm on $\mathbb{R}^d$, then you can take $G=O(d,\mathbb{Z})\ltimes\mathbb{Z}^d$ and $H=O(d,\mathbb{Z})$. [I see that S.Carnahan has posted the same suggestion right before me. Edit: and also Will Sawin]