To identify the group of an element based on ionization enthalpies, observe the jump between successive values:

1. Element X: The first ionization enthalpy ($IE_1$) is low (419 kJ mol⁻¹), but the second ionization enthalpy ($IE_2$) is extremely high (3069 kJ mol⁻¹). This huge jump indicates that removing the second electron disrupts a stable noble gas core. This is characteristic of Alkali Metals (Group 1), which have one valence electron ($ns^1$). (Note: The value 419 kJ mol⁻¹ corresponds to Potassium).
2. Element Y: The $IE_1$ (590 kJ mol⁻¹) is higher than that of X, and the $IE_2$ (1145 kJ mol⁻¹) is roughly double the first. There is no sudden massive jump comparable to X, implying the second electron is also removed from the valence shell. This behavior is characteristic of Alkaline Earth Metals (Group 2), which have two valence electrons ($ns^2$). (Note: The value 590 kJ mol⁻¹ corresponds to Calcium).