Transitive permutation groups satisfy orbit dimension σ(G) ≤ |Ω| − rank(G) + 1 with equality cases described; for the symmetric group on k-subsets this equals the metric dimension of the Johnson graph J(n,k).
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Some remarks on the orbit dimension of transitive groups and on the metric dimension of Johnson graphs
Transitive permutation groups satisfy orbit dimension σ(G) ≤ |Ω| − rank(G) + 1 with equality cases described; for the symmetric group on k-subsets this equals the metric dimension of the Johnson graph J(n,k).