The del Pezzo surface \(Y\) of degree \(5\) is a smooth nondegenerate in \(\mathbb{P}^5\), which is obtained as the blowup of the project plane in \(4\) points of which no three are collinear. Its automorphism group is the symmetric group \(\mathfrak{S}_5\). The authors give explicitly a \(6 \times 6\) antisymmetric matrix whose \(4 \times 4\)-Pfaffians are the canonical \(\mathfrak{S}_5\)-invariant Pfaffian equations of \(Y\)(Theorem 1.1). Also, the authors give concrete geometric descriptions of the irreducible representations of \(\mathfrak{S}_5\)(Section 3). Finally, the authors show that \(\mathfrak{S}_5\)-invariant equations for the embedding of \(Y\) inside \((\mathbb{P}^1)^5\) have the same Hilbert resolution as for the del Pezzo surface of degree 4 (Theorem 5.2).