Suppose I am a member of a club/group. What is that state of being inside the club called, in the following context?
I have __ to this club.
I'm thinking admission, but that really doesn't sound like it.
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The original phrasing of the question differs a lot from your suggested idea of a better term than admission in an example sentence.
'What is that state of being inside the club called...?'
The answer to 'being a member of the club' can be 'belong', as in:
I belong to this club.
(This can be a powerful statement, say, when someone questions your locus standi within the club premises.)
However, if you must use the form 'I have __ to this club.' only, then you can say:
I have access to this club.