The $beta^{-}$ decay of $^{81}$Zn to the neutron magic $N=50$ nucleus $^{81}$Ga, with only three valence protons with respect to $^{78}$Ni, was investigated. The study was performed at the ISOLDE facility at CERN by means of $gamma$ spectroscopy. The $^{81}$Zn half-life was determined to be $T_{1/2}=290(4)$ ms while the $beta$-delayed neutron emission probability was measured as $P_n=23(4)%$. The analysis of the $beta$-gated $gamma$-ray singles and $gamma$-$gamma$ coincidences from the decay of $^{81}$Zn provides 47 new levels and 70 new transitions in $^{81}$Ga. The $beta^-$$n$ decay of $^{81}$Zn was observed and a new decay scheme into the odd-odd $^{80}$Ga nucleus was established. The half-lives of the first and second excited states of $^{81}$Ga were measured via the fast-timing method using LaBr$_3$(Ce) detectors. The level scheme and transition rates are compared to large-scale shell-model calculations. The low-lying structure of $^{81}$Ga is interpreted in terms of the coupling of the three valence protons outside the doubly-magic $^{78}$Ni core.
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