Small-world networks permeate modern society. In this paper we present a methodology for creating and analyzing a practically limitless number of networks exhibiting small-world network properties. More precisely, we analyze networks whose nodes are Facebook groups sharing a common word in the group name and whose links are mutual members in any two groups. By analyzing several numerical characteristics of single networks and network aggregations, we investigate how the small-world properties scale with a coarsening of the network. We show that Facebook group networks have small average path lengths and large clustering coefficients that do not vanish with increased network size, thus exhibiting small-world features. The degree distributions cannot be characterized completely by a power law, and the clustering coefficients are significantly larger than what would be expected for random networks, while the average shortest paths have consistently small values characteristic of random graphs. At the same time, the average connectivity increases as a power of the network size, while the average clustering coefficients and average path lengths do not exhibit a clear scaling with the size of the network. Our results are somewhat similar to what has been found in previous studies of the networks of individual Facebook users.