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The particle-in-cell (PIC) algorithm is one of the most widely used algorithms in computational plasma physics. With the advent of graphical processing units (GPUs), large-scale plasma simulations on inexpensive GPU clusters are in reach. We present an implementation of a fully relativistic plasma PIC algorithm for GPUs based on the NVIDIA CUDA library. It supports a hybrid architecture consisting of single computation nodes interconnected in a standard cluster topology, with each node carrying one or more GPUs. The internode communication is realized using the message-passing interface. The simulation code PIConGPU presented in this paper is, to our knowledge, the first scalable GPU cluster implementation of the PIC algorithm in plasma physics.