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The computation of the Noether numbers of all groups of order less thanthirty-two is completed. It turns out that for these groups in non-modularcharacteristic the Noether number is attained on a multiplicity freerepresentation, it is strictly monotone on subgroups and factor groups, and itdoes not depend on the characteristic. Algorithms are developed and used todetermine the small and large Davenport constants of these groups. For each ofthese groups the Noether number is greater than the small Davenport constant,whereas the first example of a group whose Noether number exceeds the largeDavenport constant is found, answering partially a question posed byGeroldinger and Grynkiewicz.

The Noether numbers and the Davenport constants of the groups of order less than 32