Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| Fitting | #390×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2309 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D67 |
| commutator | C22×C67 | C23×C4 |
| Fitting | #390×C67 | C2 |
| socle | C23×C67 | C24 |
| hypercenter | #390 | D67 |
| hypocenter | C67 | #2309 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C23 ◁ #390 | |
1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|a^2=g,b^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce,h^d=h^66968 conjugacy classes — view the full character table.
| maximal subgroup of | #30764942, #30765220, #30770802, … |
| maximal normal subgroup of | #30764942, #30765220, #30770802, … |
| maximal quotient of | #30777068, #30777136, #30777137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 7089);