Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D257⋊M16 |
| Sylow | C257 ⋊ C2×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#11759967 |
| commutator, socle | C2×C257 | C22×C4 |
| Frattini | C2 | C22×#11759967 |
| Fitting | C2×C4×C257 | C4 |
| hypercenter | C2×C4 | #11759967 |
| hypocenter | C257 | C2×M16 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
68 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=d,d^2=e,b^a=be,f^a=f^241,f^c=f^256,f^d=f^256532 conjugacy classes — view the full character table.
| maximal subgroup of | #30657604, #30657606, #30657608, … |
| maximal normal subgroup of | #30657604, #30657606, #30657608, … |
| maximal quotient of | #30657696, C4×#16861627, C4×#16861628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 231);