Monoisotopic samples of exotic, neutron-rich 117Rh nuclei, produced in the proton-induced fission of 238U and separated using the IGISOL mass separator coupled to the JYFLTRAP Penning trap, were used to perform β and γ coincidence spectroscopy of 117Pd. The spin parity of the ground state of 117Pd was determined to be 1/2+ and the 19.1 ms isomer at 203.2 keV was assigned a spin-parity 7/2−. The 117Rh β−-decay scheme was considerably extended, and various sequences of the levels were interpreted as resulting from the prolate, oblate, and triaxial nuclear shapes. Some of the β− decays were considered as the allowed Gamow-Teller transitions. The experimental distribution of Gamow-Teller strength is discussed.