The rapid growth of fingerprint authentication-based applications makes presentation attack detection, which is the detection of fake fingerprints, become a crucial problem. There have been numerous attempts to deal with this problem; however, the existing algorithms have a significant trade-off between accuracy and computational complexity. This paper proposes a presentation attack detection method using Convolutional Neural Networks (CNN), named fPADnet (fingerprint Presentation Attack Detection network), which consists of Fire and Gram-K modules. Fire modules of fPADnet are designed following the structure of the SqueezeNet Fire module. Gram-K modules, which are derived from the Gram matrix, are used to extract texture information since texture can provide useful features in distinguishing between real and fake fingerprints. Combining Fire and Gram-K modules results in a compact and efficient network for fake fingerprint detection. Experimental results on three public databases, including LivDet 2011, 2013 and 2015, show that fPADnet can achieve an average detection error rate of 2.61%, which is comparable to the state-of-the-art accuracy, while the network size and processing time are significantly reduced.