The radical political club that dominated the French Revolution during its most intense phase was the Jacobin Club.
Founded in 1789 by deputies opposed to royal absolutism, the club was originally known as the Society of the Friends of the Constitution. It met in a former Dominican convent on Paris’s Rue Saint-Honoré, which gave its members the nickname “Jacobins.” The organization later developed a nationwide network of affiliated clubs.
The Jacobins became especially powerful after the monarchy was overthrown in 1792. The Montagnard faction, associated with figures such as Maximilien Robespierre, gained control of the revolutionary government in 1793. During this period, the Committee of Public Safety led emergency measures and repression that historians call the Reign of Terror.
The Jacobin Club was not identical to every radical group in the Revolution, and it included different factions at different times. After Robespierre’s fall on 27 July 1794, the club was closed in November 1794. The Girondins and Cordeliers were important revolutionary factions or clubs, but neither was the central radical club represented by this answer.