  We give an explicit description for a basis of a subgroup of finite index in
the group of central units of the integral group ring $\Z G$ of a finite
abelian-by-supersolvable group such that every cyclic subgroup of order not a
divisor of 4 or 6 is subnormal in $G$. The basis elements turn out to be a
natural product of conjugates of Bass units. This extends and generalizes a
result of Jespers, Parmenter and Sehgal showing that the Bass units generate a
subgroup of finite index in the center $\mathcal{Z} (\U (\Z G))$ of the unit
group $\U (\Z G)$ in case $G$ is a finite nilpotent group. Next, we give a new
construction of units that generate a subgroup of finite index in
$\mathcal{Z}(\U(\Z G))$ for all finite strongly monomial groups $G$. We call
these units generalized Bass units. Finally, we show that the commutator group
$\U(\Z G)/\U(\Z G)'$ and $\mathcal{Z}(\U(\Z G))$ have the same rank if $G$ is a
finite group such that $\Q G$ has no epimorphic image which is either a
non-commutative division algebra other than a totally definite quaternion
algebra, or a two-by-two matrix algebra over a division algebra with center
either the rationals or a quadratic imaginary extension of $\Q$. This allows us
to prove that in this case the natural images of the Bass units of $\Z G$
generate a subgroup of finite index in $\U(\Z G)/\U(\Z G)'$.
