Famous school event's outside, happening (4)
I believe the answer is:
eton
'famous school' is the definition.
(English private school)
'event's outside happening' is the wordplay.
'outside' suggests removing the centre.
'happening' becomes 'on'.
'event' with its middle removed is 'et'.
'et'+'on'='ETON'
(Other definitions for eton that I've seen before include "Posh British public school" , "Famous English public school" , "academic institution" , "Public school, established in 1440" , "Independent school" .)