\(S_8\) inside \(A_n\)

Algebra Level 5

What is the smallest positive integer \(n\) such that there is an injective homomorphism \( \phi \colon S_8 \to A_n\)?

Notation: \(S_8\) is the symmetric group on 8 symbols, and \(A_n\) is the alternating group consisting of the even permutations on \( n\) symbols.

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