From Gineipaedia, the Legend of Galactic Heroes wiki
The Republican Faction was a political coalition active in the final years of the Galactic Federation.
During the final years of the Galactic Federation, Hassan el-Sayyid led the republican faction in opposition to Rudolf von Goldenbaum's increasingly powerful fascist part, the National Reform Alliance. These efforts were ultimately unsuccessful. (Legend of Galactic Heroes Volume 1: Dawn)
the Republican Faction would have been a left-leaning party relative to the far-right NRA, and clearly valued the democratic ideals the Federation was founded on.
The pro-democracy rhetoric of the Republican Party would eventually serve as the inspiration, and perhaps model, for the Free Planets Alliance. Arguably, the FPA may have seen themselves as the direct inheritors of the Republican Faction, and therefore the only legitimate successors to the Galactic Federation--therefore providing a casus belli for their war against the Galactic Empire.