The increasing availability of sensors imaging cloud and precipitation particles, like the Multi-Angle Snowflake Camera (MASC), has resulted in datasets comprising millions of images of falling snowflakes. Automated classification is required for effective analysis of such large datasets. While supervised classification methods have been developed for this purpose in recent years, their ability to generalize is limited by the representativeness of their labeled training datasets, which are affected by the subjective judgment of the expert and require significant manual effort to derive. An alternative is unsupervised classification, which seeks to divide a dataset into distinct classes without expert-provided labels. In this paper, we introduce an unsupervised classification scheme based on a generative adversarial network (GAN) that learns to extract the key features from the snowflake images. Each image is then associated with a distribution of points in the feature space, and these distributions are used as the basis of K-medoids classification and hierarchical clustering. We found that the classification scheme is able to separate the dataset into distinct classes, each characterized by a particular size, shape and texture of the snowflake image, providing signatures of the microphysical properties of the snowflakes. This finding is supported by a comparison of the results to an existing supervised scheme. Although training the GAN is computationally intensive, the classification process proceeds directly from images to classes with minimal human intervention and therefore can be repeated for other MASC datasets with minor manual effort. As the algorithm is not specific to snowflakes, we also expect this approach to be relevant to other applications.