I wouldn't be so quick to assume that blacks couldn't vote in DC 1956. Before the Voting Rights Act, there were black voters (they were key to Truman's 1948 victory). It was only in certain locations, many in the Deep South, where blacks were stopped by local authorities. By the 1950s, even many anti-Civil Rights officials, such as Louisiana Governor Earl Long, advocating giving blacks the vote (presuming that this would mean more votes for them). Anybody know the racial makeup of DC back then?
Negro suffrage in DC was passed by Congress in 1866, before passage of either the 14th or 15th amendments. Of course, later in the 19th century, Congress eliminated municipal self-government for Washington and Georgetown (which were separate cities at that time).