In many species, cultures, and contexts, social dominance reflects the ability to exert influence over the behavior of others. Yet the behavioral attributes of those in dominant positions, and the behaviors of actually influential individuals may not be the same, and the behavioral attributes that generate influence in one social context may reduce influence in others. The question of what makes an effective leader is therefore not straightforward, and has many answers depending on the context in which leadership and influence is to be manifested. Most importantly, social dominance cannot always be assumed to be equivalent with social influence. Here we examine whether socially dominant males in the cichlid fish Astatotilapia burtoni are more effective in exerting social influence than socially subordinate males. Using machine-vision based automated tracking of behavior, we find that dominant males in this species display behavioral traits that typify leadership across taxonomic systems -they are aggressive, occupy central social network positions, and lead group movements, whereas subordinate males are passive, socially peripheral, and have little influence over typical group movement. However, in a more complex group-consensus task the influence of dominant males breaks down, and subordinate males become more effective agents of social change. In a more sophisticated group consensus task involving a visual association task, the behavioral attributes that define male dominance -aggression, rapid movement, and increased physical distance to othersinterfere with the ability of dominant males to generate group to consensus. Dominant males occupy more spatially distant positions, and had lower signal-to-noise ratio of informative behavior in the association task, while subordinate males are typically is close physical association with their group members, have high signal-to-noise behaviors in the association task, and equal visual connectivity to other group members as dominant males. The attributes that define effective social influence are therefore highly context-specific in this species. These results demonstrate that in this and many other species including humans, behavioral traits that are typical of socially dominant individuals may be the same that reduce their social influence in other contexts.