The Barth sextic is a famous example of a sextic surface with 65 nodes, the maximum number. We show that every sextic surface with 65 nodes is the discriminant of a projection from a 5-dimensional cubic hypersurface, with centre a plane contained in it. More precisely, each sextic is a discriminant in two different ways: the cubic can have 31 nodes and the plane passes through 3 nodes, or the cubic has 32 nodes and the plane passes through 2 nodes. Somewhat surprisingly, a sextic with 65 nodes cannot be the projection from a cubic with more than 32 nodes. We show that the automorphism group of the Barth sextic is the icosahedral group. Exploiting the action of this group on the set of nodes leads us to rediscover the so-called Doro-Hall graph and allows us to determine the extended code of the Barth surface. We explore the intimate connections between these objects and the geometry of the Barth sextic. Finally, we present determinantal equations for the Barth sextic, based on the results of Chap. 2 , on the knowledge of the extended code, and on a classification of half-even sets of nodes of small cardinality on sextics.