The management of the system is vested in a board of retirement, consisting of five members, one of whom shall be the county treasurer acting in an exofficio capacity. The second and third members shall be either members or pensioners of the system elected by the members within 30 days after the system becomes operative, in a manner to be determined by the board of supervisors. The fourth and fifth members shall be qualified electors of the county who are not connected with county government in any capacity, appointed by the board of supervisors within 30 days after that date, except that one of such members may be a member of the board of supervisors.