We study the time evolution of group sizes of an online community. The empirical analysis shows that (i) groups grow linearly in time, (ii) the growth values have heavy-tailed distribution and (iii) the number of new groups grows linearly with time. Based on these findings we propose a minimal model for the dynamics of elements (groups) with associated counters (sizes). We show that the elements intrinsic heterogeneity, introduced as a heavy-tailed distribution of counters growth, shapes the statistical properties of the system, such as the heavy-tailed distribution of counters and the average growth of counters proportional to the value of counters. Finally, we compare the model with a model of preferential growth and emphasize that the model based on big-gets-bigger principle lacks heterogeneity and introduces very strong correlation between size and age, whereas the model based on broad heterogeneity performs close to what is observed in the studied system.