The 51st century is the original timezone of Magnus Greel in the 1977 Doctor Who serial The Talons of Weng-Chiang. Greel was "Minister of Justice" under the tyrannical "Supreme Alliance," and responsible for the deaths of 100,000 people, earning him the epithet "the Butcher of Brisbane." In 5000, the Filipino Army defeats the Alliance at the Battle of Reykjavík, during the closing stages of World War V. Using zygma-energy powered time travel technology, Greel escapes to the 19th century Chinese Empire and steals energy from women to survive, but dies in Victorian London.
There are many references to the 51st century in the revived series of Doctor Who. One of the Doctor'scompanions, Captain Jack Harkness, originates from the 51st century, although it is unclear whether or not he or his parents participated in the events of World War V described above.
Further references in Doctor Who include the events of "The Girl in the Fireplace" partly take place during sometime in the 51st Century, on a spaceship inhabited by malfunctioning clockwork robots, who open time windows to the past. The two part story "Silence in the Library"/"Forest of the Dead" and part of the events of "The Time of Angels", each involving River Song are also set during this period, partly in a prison River Song is in. This is also the home time of K-9 Mark I. Apparently by this time humanity has spread to many worlds around the cosmos.
In the science fiction/fantasy series The Pendragon Adventure, the territory (dimension) of Third Earth exists in 5010, the future of Second Earth (and in that sense, First Earth as well, which takes place in 1937). It is presented as a utopian society, with almost every man-made structure situated underground or in space, leaving the surface of the Earth to flourish.
In the Blake and Mortimer universe, this is the last century of existence for an unnamed world totalitarian government, which originally took over in the late 21st century.
In 5060, a worldwide revolution, commanded by Focas, overthrows this regime and presumably builds a better and more just world.
In Terror from the Year 5000, a 1958 American science fiction film, a time machine made by Professor Howard Erling brings back a deformed woman from circa 5200 AD (in contradiction of the title).
In 5700, in the DC Comics universe, the Green Lantern at this point will be Pol Manning (actually an identity taken by Hal Jordan and Salakk). By now the Solar System is run by a government led by the Solar Director. The technology is capable of taking Green Lantern from his own time and implanting new memories of his identity into his mind.
In reference to this, according to Amalgam Comics continuity, the Iron Lantern at this point will be Arno Manning, defending a futuristic Coast City.