Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Sylow | C67 ⋊ #2308 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D67 |
| commutator | C22×C67 | C23×C4 |
| Fitting | C23×D4×C67 | C2 |
| socle | C23×C67 | C24 |
| hypercenter | C23×D4 | D67 |
| hypocenter | C67 | #2308 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
1160 conjugacy classes of subgroups; only the 309 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 309 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^a=be,c^a=cf,d^a=de,h^a=h^66,c^b=ce1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30749328, #30749331, #30749345, … |
| maximal normal subgroup of | #30749328, #30749331, #30749345, … |
| maximal quotient of | #30777072, #30777077, #30777133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 7088);