how did the characters of the lord of the flies abuse the powers they had?

Power is often a source of violence in Lord of the Flies. The desire for power breaks down the boundaries set by rules and order, causes strife and competition, and governs the actions of many of the boys on the island. Once achieved, power has the ability to either improve or corrupt its holder. Ralph, the more noble of the two leaders on the island, is bettered by his position as chief; whereas Jack, the usurper, abuses his power for personal gain.

You could say that both Ralph and Jack abuse their power in this book. Both of them do this quite a bit in their dealings with Piggy.

From the beginning of the book, Ralph abuses Piggy. He calls him "Piggy" even though Piggy hates the nickname. He also is constantly telling him to shut up.

While Ralph verbally abuses Piggy, Jack is also willing to abuse him physically as well. For example, in Chapter 4, Jack punches Piggy, causing his glasses to fly off and break. Later on in the story, he will even steal the remaining lens of Piggy's glasses.

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In addition to the previous post, I would like to add that Roger also abuses power in the novel. Although he is not the chief, he certainly represents the darkest side of human nature. One might compare him to military commanders in dictatorships or police states: they may not be the one with power nominally, but they surely hold the power behind the scenes. For example, when one of the lookouts tells him Jack is beating a follower, Roger muses on the possibilities of irresponsible authority. He recognizes how much freedom exists in chaos. In such an atmosphere, where there are no legal or moral restraints, he is free to act as he pleases and satisfy his sadistic tendencies. Unlike Jack, who sees violence as necessity, Roger views it as opportunity.

The twins, Samneric, recognize this too. After being beaten and forced to join Jack's tribe, they speak to Ralph, who attempts to win them over to his side again. but they are terrified.

“You don’t know Roger. He’s a terror.”

“And the chief—they’re both—”

“—terrors—”

“—only Roger—”

You can see that it is not really Jack they are afraid of: it's Roger. They confirm this with their final statement to Ralph “Roger sharpened a stick at both ends.” That one line sums up the delight Roger has discovered in abusing others. He uses the power of Jack to enact his own torture fantasies. Perhaps nowhere is this more evident than in Piggy's death. Roger watches the rock crush Piggy almost serenely, seeming to revel in his first kill. He is truly the one who reveals the beast within.