The most common and most likely answer to this crossword clue is the 1 plus 4 letter word A LIST
I've seen this crossword clue in The Times
A
L
I
S
T
More clues leading to the result A LIST
Buddy explains!

The straight part
We know that "of the highest class" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case of the highest class = a list.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.