Taxonomic hierarchy

The horse belongs to order Perissodactyla of class Mammalia. Perissodactyla includes odd-toed mammals.

Taxonomic hierarchy: Kingdom $\rightarrow$ Division/Phylum $\rightarrow$ Class $\rightarrow$ Order $\rightarrow$ Family $\rightarrow$ Genus $\rightarrow$ Species.
As we go down the hierarchy, the number of common characters increases and the category becomes less general (more specific).
Species is below genus, so it has fewer general and more specific characters. Division, class and family are above genus and are more general.
So species is less general in characters than genus.