Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C67 ⋊ #2985 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D67 |
| commutator, socle | C22×C67 | C23×C4 |
| Fitting | #432×C67 | C2 |
| hypercenter | #432 | D67 |
| hypocenter | C67 | #2985 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C4 ◁ #432 | |
322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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=f,b^2=e,c^2=e,f^2=g,b^a=be,d^a=de,h^a=h^66,c^b=ce,d^b=dg,h^b=h^66968 conjugacy classes — view the full character table.
| maximal subgroup of | #30752422, #30752426, #30752436, … |
| maximal normal subgroup of | #30752422, #30752426, #30752436, … |
| maximal quotient of | #30854586, #30854609, #30854905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 13607);