  Let B be a block of a finite group G with defect group D. We prove that the
exponent of the center of D is determined by the character table of G. In
particular, we show that D is cyclic if and only if B contains a "large" family
of irreducible p-conjugate characters. More generally, for abelian D we obtain
an explicit formula for the exponent of D in terms of character values. In
small cases even the isomorphism type of D is determined in this situation.
Moreover, it can read off from the character table whether |D/D'|=4 where D'
denotes the commutator subgroup of D. We also propose a new characterization of
nilpotent blocks in terms of the character table.
