  Recent experimental observation of magicity in $^{78}$Ni has infused the
interest to examine the persistence of the magic character across the N$=$50
shell gap in extremely neutron rich exotic nucleus $^{78}$Ni in ground as well
as excited states. A systematic study of Ni isotopes and N$=$50 isotones in
ground state is performed within the microscopic framework of relativistic
mean-field (RMF) and the triaxially deformed Nilson Strutinsky model (NSM).
Ground state density distributions, charge form factors, radii, separation
energies, pairing energies, single particle energies and the shell corrections
show strong magicity in $^{78}$Ni. Excited nuclei are treated within the
statistical theory of hot rotating nuclei where the variation of level density
parameter and entropy shows significant magicity with a deep minima at N$=$50,
which, persists up to the temperatures $\approx$ 1.5$-$2 MeV and then slowly
disappear with increasing temperature. Rotational states are evaluated and
effect of rotation on N$=$50 (Z$=$20$-$30) isotones are studied. Our results
agree very well with the available experimental data and few other theoretical
calculations.
