Power is the ability to influence others towards the attainment of specific goals, and it is a fundamental force that shapes behavior at all levels of human existence. Several theories on the nature of power in social life exist, especially in the context of social influence. Yet, in bargaining situations, surprisingly little is known about its role in shaping social preferences. Such preferences are considered to be the main explanation for observed behavior in a wide range of experimental settings. In this work, we set out to understand the role of bargaining power in the stylized environment of a Generalized Ultimatum Game (GUG). We modify the payoff structure of the standard Ultimatum Game (UG) to investigate three situations: two in which the power balance is either against the proposer or against the responder, and a balanced situation. We find that other-regarding preferences, as measured by the amount of money donated by participants, do not change with the amount of power, but power changes the offers and acceptance rates systematically. Notably, unusually high acceptance rates for lower offers were observed. This finding suggests that social preferences may be invariant to the balance of power and confirms that the role of power on human behavior deserves more attention.