Flavor changing neutral currents coming from a new nonuniversal neutral gauge boson and from the nonunitary quark mixing matrix for the $SU(3{)}_{c}\ensuremath{\bigotimes}SU(3{)}_{L}\ensuremath{\bigotimes}U(1{)}_{X}$ model with right-handed neutrinos are studied. By imposing as experimental constraints the measured values of the $3\ifmmode\times\else\texttimes\fi{}3$ quark mixing matrix, the neutral meson mixing, and bounds and measured values for direct flavor changing neutral current processes, the largest mixing of the known quarks with the exotic ones can be established, with new sources of flavor changing neutral currents being identified. Our main result is that for a $|{V}_{tb}|$ value smaller than 1, large rates of rare top decays such as $t\ensuremath{\rightarrow}c\ensuremath{\gamma}$, $t\ensuremath{\rightarrow}cZ$, and $t\ensuremath{\rightarrow}cg$ (where $g$ stands for the gluon field) are obtained; but if $|{V}_{tb}|\ensuremath{\sim}1$ the model can survive present experimental limits only if the mass of the new neutral gauge bosons becomes larger that 10 TeV.