This is the 3485th most frequent Russian word.
публичного
"Public" or "pertaining to the public."
The word 'публичного' is used here as an adjective in the genitive form modifying 'обсуждении', describing the discussion as public.
Он участвовал в публичного обсуждении проблемы.
He participated in the public discussion of the issue.
The word 'публичного' is used as an adjective in the genitive form modifying 'мероприятия', specifying the event as public.
Для проведения публичного мероприятия необходимо получить разрешение.
To hold a public event, it is necessary to obtain permission.
Here, 'публичного' acts as an adjective in the genitive form, describing 'характера' to denote something of a public nature.