Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊#14523346 |
| Sylow | C257 ⋊ C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#11759967 |
| commutator | C2×C257 | C2×C8 |
| Fitting | C23×C257 | C4 |
| socle | C22×C257 | C2×C4 |
| hypercenter | C23 | #11759967 |
| hypocenter | C257 | C22⋊C8 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C22 ◁ C23 | |
50 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=d,d^2=e,b^a=bc,f^a=f^241,f^d=f^256532 conjugacy classes — view the full character table.
| maximal subgroup of | #30657580, #30657581, #30657582, … |
| maximal normal subgroup of | #30657580, #30657581, #30657582, … |
| maximal quotient of | #30657580, #30657582, D4⋊#14523346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 202);