Texts can be sent to a group in two ways - as an individual text to each member of the group, or as a single text to all members of the group. How it's sent determines how you receive it. No matter what your phone does, if I send a group text as individual texts to each member of the group, all you'll get is a text to you, nothing about the group.
(If you're comparing this to an iPhone, iPhones don't text, they iMessage. While it may look the same to you, arsenic and sugar look the same too. SMS and iMessage are totally different under the hood.)