What are the finite groups with 8 or 16 conjugacy classes? What are the list of finite groups with 8 or 16 conjugacy classes?
I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, $|D_{13}|=26$. Or some people may denote $D_{10}$ as $D_{20}$.) Of course we have trivial examples $Z_8$ and $Z_{16}$ have  8 or 16 conjugacy classes for each.

Are there other examples of non-Abelian groups with 8 or 16 conjugacy classes? I am mostly interested in the non-Abelian groups. Thank you. :o)

Add: Partial answers are fine. (Such as answering what Jack Schmidt points out there are 18 isomorphism classes of 8 conjugacy classes.)
 A: Here are the 18 isomorphism classes of finite groups with 8 conjugacy classes:


*

*SmallGroup( 20,  1) = $\operatorname{AGL}(1,5)$

*SmallGroup( 20,  4) = $D_{10}$

*SmallGroup( 24, 13) = $C_2 \times A_4$

*SmallGroup( 26,  1) = $D_{13}$

*SmallGroup( 48,  3) = $C_3 \ltimes (C_4 \times C_4)$

*SmallGroup( 48, 28) = $\operatorname{SL}(2,3) \mathsf{Y} C_4$

*SmallGroup( 48, 29) = $\operatorname{GL}(2,3)$

*SmallGroup( 48, 50) = $C_3 \ltimes (C_2^4)$

*SmallGroup( 56, 11) = $\operatorname{AGL}(1,8)$

*SmallGroup( 68,  3) = $C_4 \ltimes C_{17}$

*SmallGroup( 78,  1) = $C_6 \ltimes C_{13}$

*SmallGroup( 80, 49) = $C_5 \ltimes C_2^4$

*SmallGroup(168, 43) = $\operatorname{A\Gamma L}(1,8)$

*SmallGroup(200, 44) = $Q_8 \ltimes (C_5 \times C_5)$

*SmallGroup(300, 23) = $C_4 \ltimes C_3 \ltimes C_5^2$

*SmallGroup(600,150) = $\operatorname{SL}(2,3) \ltimes C_5^2$

*SmallGroup(660, 13) = $\operatorname{PSL}(2,11)$

*SmallGroup(720,765) = $M_{10}$, the Mathieu group on 10 points


This list (with a computer-free proof of correctness) can be found on page 310 of (Vera-López–Vera-López, 1985).


*

*Vera López, Antonio; Vera López, Juan.
“Classification of finite groups according to the number of conjugacy classes.”
Israel J. Math. 51 (1985), no. 4, 305–338.
MR804489
DOI:10.1007/BF02764723
