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Recent development on Ge$_{1-x}$Sn$_x$ nanowires with high Sn content, beyond its solid solubility limit, make them attractive for all group-IV Si-integrated infrared photonics at nanoscale. Herein, we report a chemical vapour deposition-grown high Sn-content Ge-Ge$_{0.92}$Sn$_{0.08}$ core-shell based single nanowire photodetector operating at the optical communication wavelength of 1.55 $\mu$m. The atomic concentration of Sn in nanowires has been studied using X-ray photoelectron and Raman spectroscopy data. A metal-semiconductor-metal based single nanowire photodetector, fabricated via electron beam lithography process, exhibits significant room-temperature photoresponse even at zero bias. In addition to the high-crystalline quality and identical shell composition of the nanowire, the efficient collection of photogenerated carriers under an external electric field result in the superior responsivity and photoconductive gain as high as ~70.8 A/W and ~57, respectively at an applied bias of -1.0 V. The extra-ordinary performance of the fabricated photodetector demonstrates the potential of GeSn nanowires for future Si CMOS compatible on-chip optical communication device applications.