Mich erinnert das daran, dass ich mich einmal gefragt habe:

Which is correct: Math teacher oder math teacher?

Als mir dann in offiziellen Ausschreibungen für derartige Stellen beide Varianten in jeweils signifikanter Häufigkeit auffielen, sehe ich diese Frage in erster Linie als eine philosophische an! :-)

edit: tendenziell würde ich ein kleines "g" vorziehen, da es sich ja eigentlich nicht um einen Eigennamen handelt.

Short answer: It depends.

Longer answer:

The word "group" is a common noun and is therefore usually lowercased.

There are a few (related) reasons for capitalizing it.

(0) Common nouns are capitalized when they are part of a proper name (as in River Thames, London Bridge, and London Stock Exchange Group).

(1) A common noun is sometimes capitalized when it is considered an alternative or shortened proper name and refers to a specific person or institution (like "the Queen" when referring to Elizabeth II).

(2) It is customary for the institution itself to capitalize such short names (the Group, the University, etc.).

(3) It is customary in legal texts (etc.) to capitalize special terms, especially when they have been defined (i.e. "the Group" meaning the group that the document is about, rather than just a reference to a group mentioned in the last sentence). 

(4) Some people capitalize words that they consider to have a "special" meaning.

In your example, "the Group" refers to the London Stock Exchange Group, and the document is published by the London Stock Exchange Group; so (1) and (2) apply.

Copy editors may disapprove to a greater or lesser extent (probably more so in the US); books are more likely to lowercase (partly to avoid apparent inconsistency).

PS: Terminology varies, of course, and terms may be used more or less loosely (e.g."noun" may sometimes be used to mean a noun phrase, and "proper noun" may sometimes be used to mean a proper name), which may add to the confusion when looking at "rules".