  This note is devoted to a special Fano fourfold defined by a four-dimensional
space of skew-symmetric forms in five variables. This fourfold appears to be
closely related with the classical Segre cubic and its Cremona-Richmond
configuration of planes. Among other exceptional properties, it is
infinitesimally rigid and has Picard number six. We show how to construct it by
blow-up and contraction, starting from a configuration of five planes in a
four-dimensional quadric, compatibly with the symmetry group $S_5$. From this
construction we are able to describe the Chow ring explicitly.
