In this paper, the minimal state space realization of a fractional order transfer function is investigated in the sense of inner dimension both for commensurate and incommensurate cases. It is shown that while the traditional tools are applicable to check the minimality of a commensurate realization, there is no specific procedure to be certain about the minimality of incommensurate realizations. In this regard, the necessary and sufficient conditions are obtained for a specific class of fractional order transfer functions by which one can verify the existence of minimal realizations with inner dimension 2. Furthermore, a transformation is introduced through which all these realizations can be transformed to each other.