The class number k(G) of a group G is the number of conjugacy classes of G. In 1903, Landau proved in [Lan03] that for every n \in \mathbb{N}, there are only finitely many finite groups with exactly n conjugacy classes. The SmallClassNr package provides access to the finite groups with class number at most 14.
These groups were classified in the following papers:
k(G) \leq 5, by Miller in [Mil11] and independently by Burnside in [Bur11]
k(G) = 6,7, by Poland in [Pol68]
k(G) = 8, by Kosvintsev in [Kos74]
k(G) = 9, by Odincov and Starostin in [OS76]
k(G) = 10,11, by Vera López and Vera López in [VLVL85] (1)
k(G) = 12, by Vera López and Vera López in [VLVL86] (2)
k(G) = 13, 14, by Vera López and Sangroniz in [VLS07]
(1) In [VLVL85], three distinct groups of the form (C_5 \times C_5) \rtimes C_4 order 100 with class number 10 are given. However, only two such groups exist, being the ones with IdClassNr equal to [10,25] and [10,26].
(2) In [VLVL86], only 48 groups with class number 12 are listed. The three missing groups are provided in the appendix of [VLS07]. These are the groups with IdClassNr equal to [12,13], [12,16] and [12,39].
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