Design and simulation of a superconducting gun cavity with 31/2 cells have been studied in order to give the first push to the electron beam for the linear accelerating system at The Institute of Accelerator Technologies at Ankara University. Electrons are accelerated through the gun cavity with the help of the Radiofrequency power suppliers from cryogenic systems. Accelerating gradient should be as high as possible to accelerate electron beam inside the cavity. In this study, electron beam reaches to 9.17 MeV energy at the end of the gun cavity with the accelerating gradient; E-c=1921 MV/m. 1.3 GHz gun cavity consists of three TESLA-like shaped cells while the special designed gun-cell includes a cathode plug. Optimized important beam parameters inside the gun cavity, average beam current 3 mA, transverse emittance 2.5 mm mrad, repetition rate 30 MHz and other parameters are obtained for the SASE-FEL System. The Superfish/Poisson program is used to design each cell of the superconducting cavity. Superconducting gun cavity and Radiofrequency properties are studied by utilizing 2D Superfish/Poisson, 3D Computer Simulation Technology Microwave Studio, and 3D Computer Simulation Technology Particle Studio. Superfish/Poisson is also used to optimize the geometry of the cavity cells to get the highest accelerating gradient. The behavior of the particles along the beamline is included in this study. ASTRA Code is used to track the particles. (C) 2015 Elsevier B.V. All rights reserved.