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There are several applications that benefit from a definition of centrality which is applicable to sets of vertices, rather than individual vertices. However, existing definitions might not be able to help us in answering several network analysis questions. In this paper, we study generalizing aggregation of centralities of individual vertices, to the centrality of the set consisting of these vertices. In particular, we propose exclusive betweenness centrality, defined as the number of shortest paths passing over exactly one of the vertices in the set, and discuss how this can be useful in determining the proper center of a network. We mathematically formulate the relationship between exclusive betweenness centrality and the existing notions of set centrality, and use this relation to present an exact algorithm for computing exclusive betweenness centrality. Since it is usually practically intractable to compute exact centrality scores for large real-world networks, we also present approximate algorithms for estimating exclusive betweenness centrality. In the end, we evaluate the empirical efficiency of exclusive betweenness centrality computation over several real-world networks. Moreover, we empirically study the correlations between exclusive betweenness centrality and the existing set centrality notions.