Finite Group DatabasebetaFinite Group Databasebeta| order | 8224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | M16×C257 ⋊ C2 |
| Sylow | C257 ⋊ #104 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊D257 |
| commutator | C4×C257 | C2×C4 |
| Frattini | C2×C4 | C2×D257 |
| Fitting | M16×C257 | C2 |
| socle | C2×C257 | C22⋊C4 |
| hypercenter | M16 | D257 |
| hypocenter | C257 | #104 |
| LCS | G ▷ C4×C257 ▷ C2×C257 ▷ C257 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ M16 | |
44 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,257|a^2=d,c^2=e,d^2=e,b^a=bc,c^a=ce,c^b=ce,f^b=f^2561294 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #104×D257, C2×#22551518, #30656752, … |
| maximal normal subgroup of | #104×D257, C2×#22551518, #30656752, … |
| maximal quotient of | #30656464, #30656467, #30656472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8224, 72);