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Benefiting from the extended coverage of simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) in comparison with the conventional reflecting-only RIS, the combination of STAR-RIS and non-orthogonal multiple access (NOMA) is a promising solution to improve spectrum/energy efficiency and enhance coverage quality for future communication networks. In this letter, we concentrate on a power minimization problem for uplink STAR-RIS-aided NOMA network with multi-group detection by alternately optimizing power control and phase shift at the STAR-RIS. Particularly, an iteration method is invoked to solve the power control problem, while a block coordinate ascend (BCA) method is developed to address the phase shift optimization problem. It is demonstrated by simulation results that the proposed scheme yields better performance than several benchmarking methods.