Im new to blender and I created this group and instead of rotating around the middle of the object, it rotates around some other point and the point moves when I move the object. How can I change the pivot point to the centre of the group?
In edit mode .. not object mode ... move your mesh relative to the origin for an improvement.
in object mode use the menus in the image below. Object/Transform
Origin to Geometry
For precise control click on part of your mesh to move the 3D Cursor then use the menu option
Origin to 3D Cursor.
Try the similar menu options.
Also I believe you are talking about a Blender Mesh Object and Not a Blender Group.