Post-accelerated isomerically purified radioactive beams, available at the CERN On-Line Isotope Mass Separator facility using the resonant ionization laser technique, have been used to study the Coulomb excitation of the I(pi) = 3(-) state in (70)Cu (Z = 29, N = 41). While first results using a I(pi) = 6(-) beam were reported previously, the present complementary experiment allows us to complete the study of the multiplet of states (3(-), 4(-), 5(-), 6(-)) arising from the pi 2p(3)/(2)nu 1g(9/2) configuration. Besides the known gamma-ray transition deexciting the 4(-) state, a ground-state. ray of 511(3) keV was observed for the first time and was unambiguously associated with the 5(-) state deexcitation. This observation fixes the energy, spin, and parity of this state, completing the low-energy level scheme of (70)Cu. B(E2) values for all possible E2 transitions within the multiplet were determined. A comparison with large-scale shell model calculations using different interactions and valence spaces shows the importance of proton excitation across the Z = 28 shell gap and the role of the 2d(5/2) neutron orbital.