Siblings and twins in finite p-groups and a group identification for the groups of order $2^9$

We investigate which invariants of groups are powerful in distinguishing non-isomorphic p-groups. We introduce the notations of siblings and twins for p-groups that are difficult to distinguish and we describe the siblings and twins among the groups of small prime-power order. We then use these ideas to device an effective group identification algorithm for the $10,494,213$ groups of order $2^9$.