The Socialist Popular Alliance Party has issued a statement calling for Interior Minister Mohamed Ibrahim to be replaced due to recent security violations.
At least 49 people were killed on Saturday's third anniversary of the revolution in clashes between the security forces and opposition groups, both Islamist and non-Islamist.
The statement also condemned the "use of the state machine and its media to force one [presidential] candidate [on the country] as it harms the democratic process and takes us back to the media and police of the pre-revolution era."
The media is holding an organised campaign to defame the January 25 revolution and its figures, the statement added.
"The revolution has not yet achieved its goals and thus the movement will continue to demand bread, freedom and social justice."
Army chief Abdel-Fattah El-Sisi is being widely promoted as Egypt's next president throughout much of the state and private media. El-Sisi supporters took to the streets on Saturday, with many demanding he runs for the presidency.
Meanwhile, opponents of the army chief were attacked by security forces as well as by his civilian supporters. Over one thousand opposition protesters were arrested on Saturday alone.
The party also condemned recent terrorist attacks and violent protests, but said: "This should not be used as an excuse to make random arrests of peaceful protesters."
The party had called for and took part in a demonstration in front of the press syndicate on Saturday, which was forcefully dispersed by the security forces, leaving several dead.
The Socialist Popular Alliance party took part in the 30 June protests against Islamist president Mohamed Morsi and supported the roadmap put forth by the interim authorities after his ouster.