We consider conditional exact tests of factor effects in designed experiments for discrete response variables. Similarly to the analysis of contingency tables, Markov chain Monte Carlo methods can be used for performing exact tests, especially when large-sample approximations of the null distributions are poor and the enumeration of the conditional sample space is infeasible. To construct a connected Markov chain over the appropriate sample space, a common approach is to compute a Markov basis. Theoretically, a Markov basis can be characterized as a generator of a well-specified toric ideal in a polynomial ring and is computed by computational algebraic softwares. However, the computation of a Markov basis sometimes becomes infeasible even for problems of moderate sizes. In this paper, we obtain the closed form expression of minimal Markov bases for the main effect models of $2^{p-1}$ fractional factorial designs of resolution $p$.