Partially due to their lack of periodic structure, quasicrystals have inherently low thermal conductivity on the order of 1 - 3 W/m-K. AlPdMn quasicrystals exhibit favorable room temperature values of electrical conductivity, 500–800 (Ω-cm)-1, and thermopower, 80 μV/K, with respect to thermoelectric applications. In an effort to further increase the thermopower and hopefully minimize the thermal conductivity via phonon scattering, quartenary Al71Pd21Mn8-XReX quasicrystals were grown. X-ray data confirms that the addition of a fourth element does not alter the quasiperiodicity of the sample. Al71Pd21Mn8-XReX quasicrystals of varying Re concentration were synthesized where x had values of 0, 0.08, 0.25, 0.4, 0.8, 2, 5, 6, and 8. Both thermal and electrical transport property measurements have been performed and are reported.