HI,
Welcome to the Support Communities
I am taking this as you have an AIM valid Screen Name Logged in.
I had to look up Blast Groups.
It seems it is a "new" (don't know date) feature to AIM
It seems it works a bit like Google Groups where you set up Groups ON their web page and then access it it in different ways.
In this case AIM have allowed a AIM Login to to join the Group.
It seems the person setting up the group can add "Members"
This FAQ page seems to state if you know the Group name you can find it at AIM and Delete yourself (See bottom of page).
As it is a Groups it will tend to "Mask" the people in it so the Block thing will not work (It might if you could list thier name as the group + Screen Name)
How did they get your name ?
@mac.com and AIM registered names can use this page https://settings.aim.com/
Go to the Life Stream and Set the Privacy Setting there
Do the same in Privacy.
Double Check the email settings if it is an AIM account.
By Default your LifeStream is Public.
If you have an @Me.com name you cannot access this page.
(iChat 6 and Messages do a dual login to AIM and Me.com that then allows AIM to "See" the password which that site cannot do).
9:40 PM Tuesday; September 4, 2012
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Lion 10.7.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
"Limit the Logs to the Bits above Binary Images." No, Seriously