Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C8×C257 ⋊ C4 |
| Sylow | C257 ⋊ #107 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊#11759967 |
| commutator | C4×C257 | C2×C4 |
| Frattini | C4 | C2×#11759967 |
| Fitting | C8×C257 | C4 |
| socle | C2×C257 | C4⋊C4 |
| hypercenter | C8 | #11759967 |
| hypocenter | C257 | #107 |
| LCS | G ▷ C4×C257 ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=d,b^2=ce,c^2=e,b^a=bc,c^a=ce,f^a=f^241,f^d=f^256526 conjugacy classes — view the full character table.
| maximal subgroup of | #30657649, #30657650, #30657663, … |
| maximal normal subgroup of | #30657649, #30657650, #30657663, … |
| maximal quotient of | #30657600, #30657601, #30657611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 215);