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As a step beyond traditional personalized recommendation, group recommendation is the task of suggesting items that can satisfy a group of users. In group recommendation, the core is to design preference aggregation functions to obtain a quality summary of all group members' preferences. Such user and group preferences are commonly represented as points in the vector space (i.e., embeddings), where multiple user embeddings are compressed into one to facilitate ranking for group-item pairs. However, the resulted group representations, as points, lack adequate flexibility and capacity to account for the multi-faceted user preferences. Also, the point embedding-based preference aggregation is a less faithful reflection of a group's decision-making process, where all users have to agree on a certain value in each embedding dimension instead of a negotiable interval. In this paper, we propose a novel representation of groups via the notion of hypercubes, which are subspaces containing innumerable points in the vector space. Specifically, we design the hypercube recommender (CubeRec) to adaptively learn group hypercubes from user embeddings with minimal information loss during preference aggregation, and to leverage a revamped distance metric to measure the affinity between group hypercubes and item points. Moreover, to counteract the long-standing issue of data sparsity in group recommendation, we make full use of the geometric expressiveness of hypercubes and innovatively incorporate self-supervision by intersecting two groups. Experiments on four real-world datasets have validated the superiority of CubeRec over state-of-the-art baselines.