In recent years, free-space optical communication based on various vortex beams has gained significant attention due to its high channel capacity and low bit error rate (BER). To investigate a novel type of vortex beam (termed as gamma beam) and its application in free-space optical communication (FSO), a comprehensive analysis of its transmission performance in weak-to-strong non-Kolmogorov turbulence has been conducted for the first time. Based on the extended Rytov method, the propagation behaviors of the gamma beam via weak-to-strong non-Kolmogorov turbulent atmosphere is explored, revealing that gamma beams may outperform LG beams and HyGG beams in certain short links. Numerical calculations are performed to analyze the effects of transmission distance, rms beam radius, receiver aperture, and other parameters on the average capacity and BER. Our results are potentially significant for free-space optical communication based on orbital angular momentum.