  In this contribution we present a construction of large networks of diameter
two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley
graphs with surprisingly simple underlying groups. For several small degrees we
construct Cayley graphs of diameter two and of order greater than $\frac23$ of
Moore bound and we show that Cayley graphs of degrees
$d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest
currently known vertex-transitive graphs of diameter two.
