Instability of N$=$Z$=$28 shell closure against quadruple deformation in $^{56}$Ni

$^{56}$Ni is expected to have a doubly closed shell configuration with the magic number N$=$Z$=$28 in a simple picture. However, the observed $E$(2$^{+}_{1})$ and $B(E$2) suggest the collectivity of $^{56}$Ni and weakened N$=$Z$=$28 shell closure. Furthermore, in the low-lying states, a super deformed (SD) band with an (f$_{7/2})^{-4}$(p$_{3/2})^{4}$ configuration is experimentally observed and it shows the existence of the SD shell gap with N$=$Z$=$28. Therefore, the detailed study of the low-lying spectrum will provide us important information on the N$=$Z$=$28 magic number in the proton-rich nuclei. In this contribution we will discuss the positive-parity excited states of $^{56}$Ni and the instability of N$=$Z$=$28 shell closure on the basis of the antisymmetrized molecular dynamics calculation. It is shown that the N$=$Z$=$28 shell closure is unstable against oblate deformation and it leads to the appearance of low-lying $\beta $- and $\gamma $-bands. It is also shown that by prolate deformation the spherical N$=$Z$=$28 shell gap easily disappears and the SD shell gap appears, which generates SD bands with (f$_{7/2})^{\mathrm{-m}}$(p$_{3/2})^{\mathrm{m}}$ configurations. These two aspects of the N$=$Z$=$28 shell closure lead the coexistence of the almost spherical ground band, $\beta $ - and $\gamma $ -bands and SD bands within small excitation energies.