In this paper, we propose a deep learning (DL)-based method to automatically identify the modulations of orthogonal frequency-division multiplexing (OFDM) signals in wireless communication systems. In particular, a cost-efficient OFDM modulation classification convolutional neural network (COM-ConvNet) is principally designed with grouped convolutional layers to reduce computing complexity significantly. Remarkably, reconstructing the high-dimensional data array of OFDM signals allows our deep network to learn the underlying sample correlations within every symbol and among different symbols sufficiently. We leverage residual connection and attention connection with element-wise addition and element-wise multiplication layers in specific-designed processing blocks to enhance the pattern learning efficiency. For performance evaluation, we test the proposed method on a synthetic six-modulation OFDM signal dataset under impaired channel conditions and conduct diverse simulations, such as ablation study, parameter investigation, and complexity analysis. COM-ConvNet achieves cost efficiency (i.e., small network size and low computational cost) while maintaining an acceptable accuracy when compared with other DL models.