The $N=Z=36$ nucleus $^{72}$Kr has been studied by inelastic scattering at intermediate energies. Two targets, $^{9}$Be and $^{197}$Au, were used to extract the nuclear deformation length, $delta_text{N}$, and the reduced $E2$ transition probability, $B(E2)$. The previously unknown non-yrast $2^+$ and $4^+$ states as well as a new candidate for the octupole $3^-$ state have been observed in the scattering on the Be target and placed in the level scheme based on $gamma-gamma$ coincidences. The second $2^+$ state was also observed in the scattering on the Au target and the $B(E2;;2^+_2 rightarrow 0^+_1)$ value could be determined for the first time. Analyzing the results in terms of a two-band mixing model shows clear evidence for a oblate-prolate shape coexistence and can be explained by a shape change from an oblate ground state to prolate deformed yrast band from the first $2^+$ state. This interpretation is corroborated by beyond mean field calculations using the Gogny D1S interaction.
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