14.92 (1999)
Solved(I. D. Macdonald). Every finite $p$-group has at least $p - 1$ conjugacy classes of maximum size. I. D. Macdonald (Proc. Edinburgh Math. Soc., 26 (1983), 233–239) constructed groups of order $2^n$ for any $n \geqslant 7$ with just one conjugacy class of maximum size. Are there any examples with exactly $p - 1$ conjugacy classes of maximum size for odd $p$?
Progress
Yes, there are, for $p = 3$ (A. Jaikin-Zapirain, M. F. Newman, E. A. O'Brien, Israel J. Math., 172, no. 1 (2009), 119–123).
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