With the growing adoption of machine learning, sharing of learned models is becoming popular. However, in addition to the prediction properties the model producer aims to share, there is also a risk that the model consumer can infer other properties of the training data the model producer did not intend to share. In this paper, we focus on the inference of global properties of the training data, such as the environment in which the data was produced, or the fraction of the data that comes from a certain class, as applied to white-box Fully Connected Neural Networks (FCNNs). Because of their complexity and inscrutability, FCNNs have a particularly high risk of leaking unexpected information about their training sets; at the same time, this complexity makes extracting this information challenging. We develop techniques that reduce this complexity by noting that FCNNs are invariant under permutation of nodes in each layer. We develop our techniques using representations that capture this invariance and simplify the information extraction task. We evaluate our techniques on several synthetic and standard benchmark datasets and show that they are very effective at inferring various data properties. We also perform two case studies to demonstrate the impact of our attack. In the first case study we show that a classifier that recognizes smiling faces also leaks information about the relative attractiveness of the individuals in its training set. In the second case study we show that a classifier that recognizes Bitcoin mining from performance counters also leaks information about whether the classifier was trained on logs from machines that were patched for the Meltdown and Spectre attacks.