2 added 220 characters in body

In elementary terms, you have to analyze the following class equation $n = 1 + h_2 + ... + h_r$ where

• n is the order of the group G
• $h_k$ denotes the number of elements in the k-th conjugacy class, and $n = c_k.h_k$.

Dividing by n, you get $1 = \frac{1}{n} + \frac{1}{c_2} + ... + \frac{1}{c_r}$ which has a finite number of solutions.

Christine Ayoub in her paper On the number of conjugate classes in a group (Proc. Internat. Conf Theory of Groups Canberra 1967) has worked out this analysis for p-groups and there are probably more recent papers on this aspect, which Scott and others allude to in the comments. See for example

1. MR2557143 Keller, Thomas Michael . Lower bounds for the number of conjugacy classes of finite groups. Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 3, 567--577.

Another way of looking at your question is to see that the number of conjugacy classes is the same as the number of irreducible representations. The character table is always square. Therefore, one could ask "what are the number of irreducible characters Irr(G) in a finite group of order n?". The number of linear characters are [G:G'] where G'=commutator subgroup but the nonlinear ones are tougher and there are papers establishing various bounds for these.

1. MR2526321 (2010d:20010) Aziziheris, Kamal ; Lewis, Mark L. Counting the number of nonlinear irreducible characters of a finite group. Comm. Algebra 37 (2009), no. 5, 1572--1578.
2. MR0689258 (84d:20014) Wada, Tomoyuki . On the number of irreducible characters in a finite group. Hokkaido Math. J. 12 (1983), no. 1, 74--82.
3. MR0798751 (87a:20006) Wada, Tomoyuki . On the number of irreducible characters in a finite group. II. Hokkaido Math. J. 14 (1985), no. 2, 149--154.
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In elementary terms, you have to analyze the following class equation $n = 1 + h_2 + ... + h_r$ where

• n is the order of the group G
• $h_k$ denotes the number of elements in the k-th conjugacy class, and $n = c_k.h_k$.

Dividing by n, you get $1 = \frac{1}{n} + \frac{1}{c_2} + ... + \frac{1}{c_r}$ which has a finite number of solutions.

Christine Ayoub in her paper On the number of conjugate classes in a group (Proc. Internat. Conf Theory of Groups Canberra 1967) has worked out this analysis for p-groups and there are probably more recent papers on this aspect, which Scott and others allude to in the comments.

Another way of looking at your question is to see that the number of conjugacy classes is the same as the number of irreducible representations. The character table is always square. Therefore, one could ask "what are the number of irreducible characters Irr(G) in a finite group of order n?". The number of linear characters are [G:G'] where G'=commutator subgroup but the nonlinear ones are tougher and there are papers establishing various bounds for these.

1. MR2526321 (2010d:20010) Aziziheris, Kamal ; Lewis, Mark L. Counting the number of nonlinear irreducible characters of a finite group. Comm. Algebra 37 (2009), no. 5, 1572--1578.
2. MR0689258 (84d:20014) Wada, Tomoyuki . On the number of irreducible characters in a finite group. Hokkaido Math. J. 12 (1983), no. 1, 74--82.
3. MR0798751 (87a:20006) Wada, Tomoyuki . On the number of irreducible characters in a finite group. II. Hokkaido Math. J. 14 (1985), no. 2, 149--154.