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Traditional approaches, including heuristic and learning-based methods, fall short of addressing these complexities holistically. In response, we propose FedGCS, a novel generative client selection framework that innovatively recasts the client selection process as a generative task. Drawing inspiration from the methodologies used in large language models, FedGCS efficiently encodes abundant decision-making knowledge within a continuous representation space, enabling efficient gradient-based optimization to search for optimal client selection that will be finally output via generation. The framework comprises four steps: (1) automatic collection of diverse \"selection-score\" pair data using classical client selection methods; (2) training an encoder-evaluator-decoder framework on this data to construct a continuous representation space; (3) employing gradient-based optimization in this space for optimal client selection; (4) generating the final optimal client selection via using beam search for the well-trained decoder. FedGCS outperforms traditional methods by being more comprehensive, generalizable, and efficient, simultaneously optimizing for model performance, latency, and energy consumption. 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