= Solution
Choose a vacuum <order parameter> $\Phi_0$ and let $H$ be its <stabilizer> in $G$. Acting with $G$ produces all symmetry-related minima; two elements produce the same minimum precisely when they differ by an element of $H$. The <vacuum manifold> is therefore
$$
\boxed{M\simeq G/H.}
$$
This assumes a single transitive orbit of minima; additional independent vacuum sectors would require more than one orbit. For a local <gauge symmetry>, $G/H$ describes the vacuum directions used in finite-energy boundary conditions and defect classification, rather than distinct observable vacua after quotienting by every gauge redundancy.
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