Hercules killed Alcyoneus either when he was returning from his 10th Labor to Eurystheus (the 12 Labors-taskmaster) or during the Gigantomachy. This was the follow-up battle to the Titanomachy, fought between the Olympian gods et al. on the one side and the giant children of Gaia on the other.
In the first version, Hercules kills Alcyoneus with his bow after the giant destroyed 12 of Hercules' (probably Trojan War party's) 4-horse chariots and the men in them. In the second version, Hercules kills Alcyoneus in battle. The gods had been fighting the giants, but inverting the customary idea of omnipotent gods and frail mortals, the Olympians couldn't win without the help of one or two lesser beings. Hercules (and Dionysus) come to their aid. Hercules tries to kill Alcyoneus, but can't do it until Athena gives him a hint: Alcyoneus can't die in the land of his birth; he needs to be taken beyond the borders of Phlegra. After dragging him away, Hercules kills Alcyoneus. Then the gods get their blows in against the other giants, and finally, they let Hercules perform a mopping up operation to make sure all their victims are actually dead. It is also said that the losing giants were later placed under the earth where they make mischief causing earthquakes and volcanoes.Hercules uses a procedure similar to the one he successfully used in the Gigantomachy on Alcyoneus when he kills the giant Antaeus (in a different adventure). Antaeus' strength comes, not from his birthplace, but directly from his mother (Gaia). Hercules kills him by holding him aloft.
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