The most common and most likely answer to this crossword clue is the 8 letter word NOBILITY
I've seen this crossword clue in The Guardian
N
O
B
I
L
I
T
Y
More clues leading to the result NOBILITY
- Lack of talent not a feature of Lord’s
- Lack of talent, after alto leaves High Society
- Lack of talent excluding a privileged group
- Lords taking in old boy I fired in US city
- Aristocrats showing complete lack of talent leaving area
- No power without a peerage
- The gentry and the celebrity dismissed the volunteers
Buddy explains!

The straight part
We know that "privileged group" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case privileged group = nobility.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.
What does NOBILITY mean?
NOBILITY noun- A noble or privileged social class, historically accompanied by a hereditary title; aristocracy.
- The quality of being noble.