  The computation of the Noether numbers of all groups of order less than
thirty-two is completed. It turns out that for these groups in non-modular
characteristic the Noether number is attained on a multiplicity free
representation, it is strictly monotone on subgroups and factor groups, and it
does not depend on the characteristic. Algorithms are developed and used to
determine the small and large Davenport constants of these groups. For each of
these groups the Noether number is greater than the small Davenport constant,
whereas the first example of a group whose Noether number exceeds the large
Davenport constant is found, answering partially a question posed by
Geroldinger and Grynkiewicz.
