The group refused to leave the grounds when told to do so, allegedly blocking prison staff from entering and leaving, and threatening police officers, the Metropolitan Police said.
The Met said several of them also managed to get inside a staff entrance area of a prison building.
The group have been arrested on suspicion of aggravated trespass, the force said.
Footage shared on social media showed a group chanting outside the prison, with some holding signs.
At one point, the group can be seen moving towards a building – and one clip appears to show some demonstrators inside.
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It is understood that they were holding the demonstration in support of Umer Khalid.
Mr Khalid is among five people accused of breaking into RAF Brize Norton last July and damaging two Voyager aircraft.
He denies the charges and has escalated his protest to a thirst strike.
