Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C257 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#11759967 |
| commutator, socle | C2×C257 | C22×C4 |
| Fitting | D4×C257 | C4 |
| hypercenter | D4 | #11759967 |
| hypocenter | C257 | #131 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C2 ◁ D4 | |
62 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=d,d^2=e,f^a=f^241,c^b=ce,f^d=f^256340 conjugacy classes — view the full character table.
| maximal subgroup of | #30657779, #30657780, #30657782, … |
| maximal normal subgroup of | #30657779, #30657780, #30657782, … |
| maximal quotient of | #30657701, #30657702, #30657710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 233);