Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including clustering, graph classification, and graph neural networks. The definition of persistent homology for graphs is relatively straightforward, as graphs possess distinct intrinsic distances and a simplicial complex structure. However, hypergraphs present a challenge in preserving topological information since they may not have a simplicial complex structure. In this paper, we define several topological characterizations of hypergraphs in defining hypergraph persistent homology to prioritize different higher-order structures within hypergraphs. We further use these persistent homology filtrations in classifying four different real-world hypergraphs and compare their performance to the state-of-the-art graph neural network models. Experimental results demonstrate that persistent homology filtrations are effective in classifying hypergraphs and outperform the baseline models. To the best of our knowledge, this study represents the first systematic attempt to tackle the hypergraph classification problem using persistent homology.