The main problem for generating an extractive automatic text summary (EATS) is to detect the key themes of a text. For this task, unsupervised approaches cluster the sentences of the original text to find the key sentences that take part in an automatic summary. The quality of an automatic summary is evaluated using similarity metrics with human-made summaries. However, the relationship between the quality of the human-made summaries and the internal quality of the clustering is unclear. First, this paper proposes a comparison of the correlation of the quality of a human-made summary to the internal quality of the clustering validation index for finding the best correlation with a clustering validation index. Second, in this paper, an evolutionary method based on the best above internal clustering validation index for an automatic text summarization task is proposed. Our proposed unsupervised method for EATS has the advantage of not requiring information regarding the specific classes or themes of a text, and is therefore domain- and language-independent. The high results obtained by our method, using the most-competitive standard collection for EATS, prove that our method maintains a high correlation with human-made summaries, meeting the specific features of the groups, for example, compaction, separation, distribution, and density.