In this article we construct for any prime power q and odd n ≥ 5, a new Fq2n -maximal curve Xn. Like the Garcia–G¨uneri–Stichtenoth maximal curves, our curves generalize the Giulietti–Korchmaros maximal curve, though in a different way. We compute the full automorphism group of Xn, yielding that it has precisely q(q2 − 1)(qn + 1) automorphisms. Further, we show that unless q = 2, the curve Xn is not a Galois subcover of the Hermitian curve. Finally, up to our knowledge, we find new values of the genus spectrum of Fq2n -maximal curves, by considering some Galois subcovers of Xn.