A study on the dielectric characteristics of the Glass Fiber Reinforced Plastic (GFRP) is important for designing a reliable high voltage superconducting machines such as transmission superconducting fault current limiters, superconducting cables, and superconducting transformers. In this paper, dielectric experiments of the GFRP under lightning impulse voltage are conducted in liquid nitrogen($LN_2$) according to various experimental conditions such as the thicknesses of the GFRP, the diameters of electrode systems and the pressures. The dielectric characteristics of the GFRP are analyzed by using a Finite Elements Method(FEM) according to various field utilization factors. It has been reported that the electrical insulation design of the GFRP would be conducted by considering the mean electric field intensity($E_{mean}$) distributed inside the GFRP. In this study, it is found that the dielectric performance of the GFRP could be explained by not only $E_{mean}$ but also the maximum electric field intensity ($E_{max}$). Finally, the empirical formulae of the GFRP to estimate an electrical breakdown voltage at sparkover under the lightning impulse condition are deduced. It is expected that the presented experimental results in this paper are helpful to design electrically reliable high voltage superconducting machines using the GFRP as an insulation material.