The olfactory system faces the difficult task of identifying an enormous variety of odors independent of their intensity. Primacy coding, where the odor identity is encoded by the receptor types that respond earliest, might provide a compact and informative representation that can be interpreted efficiently by the brain. In this paper, we analyze the information transmitted by a simple model of primacy coding using numerical simulations and statistical descriptions. We show that the encoded information depends strongly on the number of receptor types included in the primacy representation, but only weakly on the size of the receptor repertoire. The representation is independent of the odor intensity and the transmitted information is useful to perform typical olfactory tasks with close to experimentally measured performance. Interestingly, we find situations in which a smaller receptor repertoire is advantageous for discriminating odors. The model also suggests that overly sensitive receptor types could dominate the entire response and make the whole array useless, which allows us to predict how receptor arrays need to adapt to stay useful during environmental changes. Taken together, we show that primacy coding is more useful than simple binary and normalized coding, essentially because the sparsity of the odor representations is independent of the odor statistics, in contrast to the alternatives. Primacy coding thus provides an efficient odor representation that is independent of the odor intensity and might thus help to identify odors in the olfactory cortex.