  Given a countable group $X$ we study the algebraic structure of its
superextension $\lambda(X)$. This is a right-topological semigroup consisting
of all maximal linked systems on $X$ endowed with the operation $$\mathcal
A\circ\mathcal B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\mathcal A\}$$
that extends the group operation of $X$. We show that the subsemigroup
$\lambda^\circ(X)$ of free maximal linked systems contains an open dense subset
of right cancelable elements. Also we prove that the topological center of
$\lambda(X)$ coincides with the subsemigroup $\lambda^\bullet(X)$ of all
maximal linked systems with finite support. This result is applied to show that
the algebraic center of $\lambda(X)$ coincides with the algebraic center of $X$
provided $X$ is countably infinite. On the other hand, for finite groups $X$ of
order $3\le|X|\le5$ the algebraic center of $\lambda(X)$ is strictly larger
than the algebraic center of $X$.
