For Graph Neural Networks, oversmoothing denotes the homogenization of vertex embeddings as the number of layers increases. To better understand this phenomenon, we study community detection with a linearized Graph Convolutional Network on the Contextual Stochastic Block Model. We express the distribution of the embeddings in each community as a Gaussian mixture over a low-dimensional latent space, with explicit formulas in the case of a single layer. This yields tractable estimators for classification accuracy at finite depth. Numerical experiments suggest that modeling with a single Gaussian is insufficient and that the impact of depth may be more complex than previously anticipated.