We report on a very low temperature heat capacity investigation of an icosahedral Al-Pd-Re sample which, from resistivity measurements, is believed to lie on the insulating side of a metal-insulator (M-I) transition. Our main results are: (i) The electronic contribution, which possibly obeys a γT variation above 1K, follows a power T a law (a~0.1) below 1K, in agreement with early results obtained for systems crossing the M-I transition. Here T denotes the absolute temperature. (ii) The large nuclear hyperfine contribution of quadrupolar origin, dominant below 300mK, can only be due to the Re nuclei. We reach this conclusion from a comparative study of icosahedral Al-Cu-Fe and Al-Pd-Mn quasicrystals. In addition, from the study of the thermal relaxation of the sample after a heat pulse, we are able to estimate the relaxation times T 1 of the different kinds of nuclei, for those which relax during our experimental time span. (iii) We finally propose a microscopic origin of the quadrupolar-electric field-gradient on the nuclei in terms of local electronic density and atomic structure.