  Compactifications of the CHL string to eight dimensions can be characterized
by embeddings of root lattices into the rank 12 momentum lattice $\Lambda_M$,
the so-called Mikhailov lattice. Based on this data, we devise a method to
determine the global gauge group structure including all $U(1)$ factors. The
key observation is that, while the physical states correspond to vectors in the
momentum lattice, the gauge group topology is encoded in its dual. Interpreting
a non-trivial $\pi_1(G) \equiv {\cal Z}$ for the non-Abelian gauge group $G$ as
having gauged a ${\cal Z}$ 1-form symmetry, we also prove that all CHL gauge
groups are free of a certain anomaly (arXiv:2008.10605) that would obstruct
this gauging. We verify this by explicitly computing ${\cal Z}$ for all 8d CHL
vacua with rank$(G)=10$. Since our method applies also to $T^2$
compactifications of heterotic strings, we further establish a map that
determines any CHL gauge group topology from that of a "parent" heterotic
model.
