In animal societies, conflicts can be resolved by combatants or through third-party intervention. In gregarious species, conflicts among pairs can spread to involve multiple individuals. In the case of large conflicts, containment and termination of aggression by third parties is important. Successful intervention relies on consensus among combatants about the intervener's capacity to use force. We refer to this consensus as power. We measure it and study how it arises, using as our model system a pigtailed macaque (Macaca nemestrina) society. In macaques, the degree to which one individual perceives another as capable of using force is communicated using a special dominance signal. Group consensus about an individual's capacity to use force arises from the network of signaling interactions. We derive a formalism to quantify consensus in the network. We find that the power distribution is fat tailed and power is a strong predictor of social variables including request for support, intervention cost, and intensity. We develop models to show how dominance-signaling strategies promote robust power distributions despite individual signaling errors. We suggest that when considering correlated interactions among many individuals it can be more useful to emphasize coarse-grained information stored at the group level--behavioral macrostates--over detailed information at the individual level.