Let $G$ be a non abelian simple group and $x,~y$ be two elements of order $p$ of $G$, where $p$ is a prime. Suppose $|x^G|\neq |y^G|$. Is there any relation between $|x^G|$ and $|y^G|$? For example is it true that $|x^G|$ divides $|y^G|$ or vice versa?
$x^G$ is the conjugacy class of $x$ in $G$, i.e. $~x^G=\{g^{-1}xg:~ g\in G\}$.
In the alternating group $G$ of degree 9, there are two conjugacy classes of elements of order $2$: $x=(1,2)(3,4)$ and $y=(1,2)(3,4)(5,6)(7,8)$. Neither class size divides the other: $|x^G| = 2\cdot 3^3 \cdot 7$ and $|y^G|=3^3\cdot 5 \cdot 7$; the first is divisible by 2 but the second is not, and the second is divisible by 5 but the first is not. $p=3$ also works for this group.
Here are the small examples, up to $|G| \leq 10^{10}$. “leftover” is a semi-colon separated list of class sizes divided by the GCD of the class sizes (given in the previous column). $$\begin{array}{cccl} G & p & \gcd & \textrm{leftover} \\ \hline M_{12} & 2 & 3^2\cdot11 & 2^2; 5 \\ M_{12} & 3 & 2^4\cdot5\cdot11 & 2; 3 \\ A_9 & 2 & 3^3\cdot7 & 2; 5 \\ A_9 & 3 & 2^3\cdot7 & 3; 2^2\cdot3\cdot5; 2^3\cdot5 \\ PSp_6(2) & 3 & 2^5\cdot7 & 3; 2^2\cdot3\cdot5; 2\cdot5 \\ A_{10} & 2 & 3^2\cdot5\cdot7 & 2; 3\cdot5 \\ A_{10} & 3 & 2^4\cdot5 & 3; 3\cdot5\cdot7; 2^3\cdot5\cdot7 \\ PSU_3(8) & 3 & 2^6\cdot19 & 3; 3; 2^3\cdot7 \\ PSL_4(3) & 2 & 3^4\cdot13 & 5; 2 \\ PSL_5(2) & 3 & 2^7\cdot31 & 5; 2\cdot7 \\ {}^2F_4(2) & 2 & 3^2\cdot5\cdot13 & 2^2\cdot5; 3 \\ A_{11} & 2 & 3^2\cdot5\cdot11 & 2; 5\cdot7 \\ A_{11} & 3 & 2\cdot5\cdot11 & 3; 2^3\cdot3\cdot7; 2^5\cdot5\cdot7 \\ HS & 2 & 5^2\cdot7\cdot11 & 3; 2^3 \\ HS & 5 & 2^7\cdot3\cdot7\cdot11 & 3; 5; 2^2\cdot3\cdot5 \\ J_3 & 3 & 2^4\cdot17\cdot19 & 3^2; 2^3\cdot5 \\ \Omega^+_8(2) & 2 & 3^2\cdot5\cdot7 & 2^2\cdot3^2\cdot5; 5; 2^2\cdot3; 2^2\cdot3; 2^2\cdot3 \\ \Omega^-_8(2) & 3 & 2^6\cdot17 & 3; 2^3\cdot3\cdot7; 2^3\cdot5\cdot7 \\ {}^3D_4(2) & 3 & 2^9\cdot7\cdot13 & 7; 3 \\ M_{24} & 2 & 3^2\cdot11\cdot23 & 5; 2\cdot7 \\ M_{24} & 3 & 2^7\cdot11\cdot23 & 3\cdot5; 7 \\ PSL_4(4) & 3 & 2^6\cdot17 & 5; 5; 2^2\cdot3\cdot7; 2^4\cdot3\cdot5\cdot7 \\ PSL_4(4) & 5 & 2^8\cdot3^2\cdot7 & 2^2\cdot17; 2^2\cdot17; 2^4\cdot3\cdot17; 3; 3 \\ PSU_4(4) & 3 & 2^8\cdot5\cdot13 & 2^2\cdot17; 5 \\ PSU_4(4) & 5 & 2^6\cdot17 & 2^4\cdot3\cdot13 ~ (6 \textrm{ classes}); \\ & & & 3 ~ (4 \textrm{ classes}); \\ & & & 2^2\cdot13 ~ (2 \textrm{ classes}); \\ & & & 2^6\cdot3^2\cdot13 \\ PSU_3(17) & 3 & 7\cdot13\cdot17^2 & 3; 3; 2^4\cdot17 \\ He & 2 & 3\cdot5\cdot7^2\cdot17 & 2; 3\cdot5 \\ He & 7 & 2^7\cdot3^2\cdot5^2\cdot17 & 2^2\cdot3\cdot7; 2^2\cdot3\cdot7; 2^3; 7; 7 \\ PSp_6(3) & 2 & 3^4\cdot13 & 7; 2^2\cdot3^2\cdot5 \\ \Omega_7(3) & 3 & 2^3\cdot7\cdot13 & 2\cdot5; \\ &&& 2^2\cdot3^2; \\ &&& 2^3\cdot3^2\cdot5 ~(2\textrm{ classes}); \\ &&& 2^4\cdot3^3\cdot5; \\ &&& 2^5\cdot3^3\cdot5 \\ &&& 3^2\cdot5; \\ PSL_3(19) & 3 & 19^2\cdot127 & 2^2\cdot5\cdot19; 3; 3 \\ PSL_4(5) & 3 & 2^3\cdot5^4\cdot31 & 5\cdot13; 2 \\ PSU_6(2) & 3 & 2^9\cdot7\cdot11 & 5; 3; 2^3\cdot3\cdot5 \\ \end{array}$$