Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊#14523348 |
| unambiguous semidirect product | (C2×#14523348)⋊C2 |
| Fitting | C22×C257 ⋊ C8 |
| Sylow | C257 ⋊ C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#14523348 |
| commutator, socle | C2×C257 | C2×C8 |
| Fitting | C22×C257 | C8 |
| hypercenter | C22 | #14523348 |
| hypocenter | C257 | C22⋊C8 |
| LCS | G ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C2 ◁ C22 | |
50 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=d,d^2=e,b^a=bc,f^a=f^64,f^d=f^241,f^e=f^256148 conjugacy classes — view the full character table.
| maximal subgroup of | #30657822, #30657823, C2×C22⋊#14523348, … |
| maximal normal subgroup of | #30657822, #30657823, C2×C22⋊#14523348, … |
| maximal quotient of | #30657822, #30657823, D4⋊#14523348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 238);