An X-band coaxial multi-beam relativistic klystron amplifier is designed in order to increase the output microwave power and the operating frequency of the amplifier tube. Based on the results from PIC simulation, the experiment is performed on the long pulse accelerator. In the experiment, the transmittal and bunching of electron beam are analyzed, and then the amplifier is driven by a beam of 4.2 kA at 700 kV, and an external source injects microwave power of 60 kW into an input cavity, then 670 MW microwave power of 5 Hz repeating frequency is extracted, the amplifier gain is about 40 dB, the pulse width is 89 ns, and the conversion efficiency is 23%. The experiment proves that several tens kW-level input power can drive the X-band coaxial multi-beam relativistic klystron amplifier to generate several hundred MW-level long pulse output power.