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Community detection is of great significance for understanding network functions and behaviors. With the successful application of deep learning in network-based analyses, recent studies have turned to utilizing graph convolutional networks (GCNs) to this problem due to their capability in capturing network attributes. Nevertheless, most existing GCN-based community detection approaches are semi-supervised and local structure-aware, even though community detection is an unsupervised learning problem essentially. In this paper, we develop a novel GCN method for unsupervised community detection under the framework of mutual information (MI) maximization, called UCDMI. Specifically, a novel MI maximization mechanism is developed to capture more fine-grained information of the global network structure in an unsupervised manner. Moreover, a new aggregation function is proposed for GCN to distinguish the importance between different neighboring nodes, which enables our method to identify more high-quality node representations and improve the community detection performance. Our extensive experiments demonstrate the effectiveness of our proposed UCDMI compared with several state-of-the-art community detection methods.