I was playing around with the number of conjugacy classes of $p$-groups in GAP and made the following conjecture:
If there is a group of order $p^{2n}$ with $k$ conjugacy classes then there is a group of order $p^{2n+1}$ with $k+p-1$ conjugacy classes. The only exception is $p=2$, $n=2$, $k=7$.
For example, there are groups of order $64$ with $\{13,16,19,22,25,28,34,40,64\}$ many conjugacy classes so there are groups of order $128$ with $\{14,17,20,23,26,29,35,41,65\}$ many conjugacy classes.
Can someone prove this or find a counterexample?
For reference, here is the data that I was using.
[1] [2] [4] [5.8] [7.10,16] [11,14,17,20,32] [13,16,19,22,25,28,34,40,64] [14,17,20,23,26,29,32,35,38,41,44,50,56,65,68,80,128] [19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,82,85,88,100,112,130,136,160,256]
[1] [3] [9] [11,27] [17,33,81] [19,35,51,83,99,243] [41,57,73,89,105,121,153,249,297,729] [43,59,75,91,107,123,139,155,171,187,219,251,267,299,315,363,459,731,747,891,2187]
[1] [5] [25] [29,125] [49,145,625] [53,149,245,629,725,3125] [73,169,265,361,649,745,841,1225,3145,3625,15625]
[1] [7] [49] [55,343] [97,385,2401] [103,391,679,2407,2695,16807]
n:=16;
l:=[];
for i in [1..NrSmallGroups(n)] do
G:=SmallGroup(n,i);
if not \in(NrConjugacyClasses(G),l) then
Add(l,NrConjugacyClasses(G));
Sort(l);
fi;
od;
Print(l);
One well-known pattern in the above data is a result of Hall that if $n=2k+e$ (with $e=0,1$) then $k=p^e+(p^2-1)(n+(p-1)t)$ for some $t\geq0$. This explains why the above lists jump by $(2-1)(2^2-1)=3$ when $p=2$, by $(3-1)(3^2-1)=16$ when $p=3$, by $(5-1)(5^2-1)=96$ when $p=5$, and by $(7-1)(7^2-1)=288$ when $p=7$.