  The effects of an extra $Z'$ gauge boson with family nonuniversal fermion
couplings on the rare top quark decay $t --> c $gamma$ are first examined in a
model independent way and then in the minimal 331 model. It is found that the
respective branching fraction is at most of the order of $10^{-8}$ for
$m_{Z'}=500$ GeV and dramatically decreases for a heavier $Z'$ boson. This
results is in sharp contrast with a previous evaluation of this decay in the
context of topcolor assisted technicolor models, which found that $B(t --> c
\gamma)\sim 10^{-6}$ for $m_{Z'}=1$ TeV.
