  Let $G$ be a group. Two elements $x, y$ are said to be {\it $z$-equivalent}
if their centralizers are conjugate in $G$. The class equation of $G$ is the
partition of $G$ into conjugacy classes. Further decomposition of conjugacy
classes into $z$-classes provides an important information about the internal
structure of the group.
  Let $I(\H^n)$ denote the group of isometries of the hyperbolic $n$-space. We
show that the number of $z$-classes in $I(\H^n)$ is finite. We actually compute
their number, cf. theorem 1.3. We interpret the finiteness of $z$-classes as
accounting for the finiteness of "dynamical types" in $I(\H^n)$. Along the way
we also parametrize conjugacy classes. We mainly use the linear model for the
hyperbolic space for this purpose. This description of parametrizing conjugacy
classes appears to be new.
