Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×Q8×C257 ⋊ C2 |
| Sylow | C257 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C22⋊D257 |
| commutator | C4×C257 | C23 |
| Frattini | C4 | C22×D257 |
| Fitting | C2×Q8×C257 | C2 |
| socle | C2×C257 | C2×D4 |
| hypercenter | C2×Q8 | D257 |
| hypocenter | C257 | #137 |
| LCS | G ▷ C4×C257 ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×Q8 | |
60 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|b^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^256,d^b=de1291 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30656913, #30656914, #30656918, … |
| maximal normal subgroup of | #30656913, #30656914, #30656918, … |
| maximal quotient of | #30656736, #30656746, #30656777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 177);