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From what I understand, hydrogen has relatively high electronegativity, especially compared to the other elements in its group, due to only having one electron in its 1s shell. Atoms are more stable and "happiest" with a full shell, so hydrogen has a higher tendency to attract an electron to itself (electronegativity), since it only needs one more electron to fill that 1s shell.
Because Hydrogen does not take an octet (as mentioned above) and it forms a 1+ ion (thus taking a max of 2 valence electrons), Hydrogen will be more willing to complete its unstable 1s shell rather than lose it (because it can't otherwise it wouldn't be an element). And because of this character, it will have a hight electronegativity.
In addition, hydrogen is a very small atom, so its nuclear charge has a strong pull on the electrons in a bond. There is also very little to no shielding because hydrogen only has one electron, so the effective nuclear charge felt by the electron is very high.
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