Surely that has finished by now.
You can find a little more from what seems to be an FOI request of 28 June 2016
The GREAT Club pilot remains in operation whilst longer term options are
considered. It currently has 83 members.
a three year old article published by the previous government which discusses stuff everybody already knows about.
Indeed! Some of the information in the FOI response is simply repeated from the initial pilot announcement.
The GREAT Club pilot - an invitation-only service to provide top business executives with bespoke support from UK Visas and Immigration
The GREAT Club offers a personalised service for a small number of the very highest-value customers who use the visa service and have strong links to the UK.
Members are provided with an account manager to ensure their journey through the visa and immigration service is swift and smooth. The account manager is also able to arrange visa services tailored to each individual’s needs at no extra cost during the pilot.
The operating costs of the GREAT Club are as set out below:
2016: £1,439.66 (as at 31 March 2016)
How does an Indian national get an invitation to join the "Great Club"?
It isn't explained who issues the invitations or what process is used to identify individuals to receive invitations. The specific FOI request didn't seem to have asked about that.
Since the club has a cap of 100 people worldwide and there are 83 members, It would seem there are only 17 vacancies worldwide.
I would suspect that if you, as an individual, are not a billionaire with vast investments in the UK and do not already have good contacts in the UK government or Foreign Office, you probably are unlikely to receive an invitation or be able to solicit one. I imagine you need to be as well connected as I expect Ratan Tata to be - but this is only an uneducated opinion from someone with no experience of such things.
Does the club offer the Standard Visitor Visa? Or some other type of visitor visa that nobody knows about?
From what I have read, there is no new type of visa, just a bit of inexpensive hand-holding to guide a few top business leaders through the expedited process.