  We show that the twisted group C$^*$-algebra associated with a discrete
FC-hypercentral group is simple (resp. has a unique tracial state) if and only
if Kleppner's condition is satisfied. This generalizes a result of J. Packer
for countable nilpotent groups. We also consider a larger class of groups, for
which we can show that the corresponding reduced twisted group C$^*$-algebras
have a unique tracial state if and only if Kleppner's condition holds.

Operator Algebras;Group Theory