  Given a group $X$ we study the algebraic structure of the compact
right-topological semigroup $\lambda(X)$ consisting of maximal linked systems
on $X$. This semigroup contains the semigroup $\beta(X)$ of ultrafilters as a
closed subsemigroup. We construct a faithful representation of the semigroup
$\lambda(X)$ in the semigroup of all self-maps of the power-set of $X$ and
using this representation describe the structure of minimal ideal and minimal
left ideals of $\lambda(X)$ for each twinic group $X$. The class of twinic
groups includes all amenable groups and all groups with periodic commutators
but does not include the free group with two generators.
