Are λίγο από connected here? I.e. does this sentence mean that they read a small part of the book. Or λίγο is independent adverb and denotes time duration so it means that they read just for a while? And if that's the case, could we place λίγο somewhere else in the sentence? At the end maybe?
Interesting that we only seem to use genitive case when using του, but it disappears in από το. I guess in this situation the βιβλιο is demoted to an indirect dative sort of object.
Because genitive wouldn't work here, especially with λίγο από. You could say Διαβάσαμε ένα μικρό μέρος του βιβλίου, if the English sentence matched that context, but not here. This is a translation that's not word-for-word.