I am trying to understand the concept of locally finitely presentable categories. I have discovered the concept of compact object here. I have discovered that for groups, the finitely presented groups are precisely the compact objects in Grp (the category of groups). I have also discovered that Grp is locally finitely presentable. This means that Grp contains objects that are not compact. Compact objects are also referred to as finitely presentable objects. I am assuming that I am confusing the terminology. Basically, how can Grp be locally finitely presentable, if it contains objects that are not finitely presentable?
Here is a link that states the Grp is locally finitely presentable. It is stated in example 4.