Outdid graduate gaining first in sociology, note (6)
I believe the answer is:
bested
'outdid' is the definition.
('best' can be a synonym of 'outdo')
'graduate gaining first in sociology note' is the wordplay.
'graduate' becomes 'bed' (I've seen this in other clues).
'gaining' is an insertion indicator.
'first in sociology' becomes 's' (1st letter of 'sociology').
'note' becomes 'te' (musical note as in do re mi fa so la ti).
's'+'te'='ste'
'bed' enclosing 'ste' is 'BESTED'.
(Other definitions for bested that I've seen before include "Defeated or outwitted" , "Worsted" , "Conquered, overcame" .)