Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C16×C257 ⋊ C2 |
| Sylow | C257 ⋊ SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D2056 |
| commutator | C8×C257 | C22 |
| Frattini | C8 | C2×D257 |
| Fitting | C16×C257 | C2 |
| socle | C2×C257 | D8 |
| hypercenter | C16 | D257 |
| hypocenter | C257 | SD32 |
| LCS | G ▷ C8×C257 ▷ C4×C257 ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
26 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^256,c^b=cd,d^b=de,f^b=f^2562059 conjugacy classes. The character table is too large to display.
| maximal subgroup of | SD32×D257, C2×#22551537, #30657168, … |
| maximal normal subgroup of | SD32×D257, C2×#22551537, #30657168, … |
| maximal quotient of | #30656539, #30656542, #30656565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 91);