Number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size
From Groupprops
Statement
Suppose is a natural number. Then, there exists a PORC function of degree such that, for any prime power , the number of conjugacy classes in the projective special linear group (i.e., the projective special linear group of degree over the finite field of size ) is .
A PORC function is a polynomial on residue classes  it looks like different polynomial functions on different congruence classes modulo a particular number. In this case, we only need to consider congruence classes modulo to define the PORC function. In fact, for a field size of , the polynomial depends only on the value .
General observations
Below, we list some general observations about the PORC function in giving number of conjugacy classes in .
Item  Value 

Degree of polynomial  for all the polynomials 
Leading coefficient of polynomial  The leading coefficient of the polynomial for is the reciprocal of 
Factors of polynomial  There are no common factors to all the polynomials 
Coefficients of polynomial  It seems like they are all integer multiples of the leading coefficient 
Comparison between polynomials  Suppose are prime powers such that divides . Then, the polynomial that works on the residue class of is bigger than the polynomial that works on the residue class of , where bigger means that the difference has positive leading coefficient. In particular, the smallest polynomial corresponds to congruence classes that are 1 more than classes relatively prime to , and the largest polynomial corresponds to the congruence class 1 mod . NOTE: This behavior is opposite to that of the PORC functions for and . However, if we consider the polynomials obtained by dividing by the leading coefficient to make the polynomials monic, and then compare, the behavior reverses and becomes similar to that for and . 
Particular cases
(degree of projective special linear group, also modulus to which we need to consider congruence classes)  (degree of PORC function)  Possibilities for  Corresponding congruence classes mod for  Corresponding polynomials in PORC function of giving number of conjugacy classes in  More information 

1  0  1  1  1  the group is a trivial group 
2  1  1 2 
0 1 

element structure of projective special linear group of degree two over a finite field 
3  2  1 3 
0 or 2 1 

element structure of projective special linear group of degree three over a finite field 
4  3  1 2 4 
0 or 2 3 1 

element structure of projective special linear group of degree four over a finite field 
Related facts
 Number of conjugacy classes in general linear group of fixed degree over a finite field is polynomial function of field size
 Number of conjugacy classes in projective general linear group of fixed degree over a finite field is PORC function of field size
 Number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size