Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C22⋊C4×C257 ⋊ C2 |
| Sylow | C257 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D257 |
| commutator, socle | C22×C257 | C23 |
| Fitting | C22⋊C4×C257 | C2 |
| hypercenter | C22⋊C4 | D257 |
| hypocenter | C257 | #123 |
| LCS | G ▷ C22×C257 ▷ C257 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
78 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|c^2=d,b^a=bd,c^a=ce,f^a=f^256,f^c=f^2561294 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30656928, #123×D257, C2×#22551576, … |
| maximal normal subgroup of | #30656928, #123×D257, C2×#22551576, … |
| maximal quotient of | #30656598, #30656612, #30656614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 130);