Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C67 ⋊ #1833 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #14523999 |
| commutator | C2×C4×C67 | C22×C4 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #344×C67 | C2 |
| socle | C2×C67 | #382 |
| hypercenter | #344 | D67 |
| hypocenter | C67 | #1833 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #344 | |
172 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 73 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=d,e^2=g,b^a=bd,c^a=ce,d^a=dg,f^b=fg,f^c=fg,h^c=h^66758 conjugacy classes — view the full character table.
| maximal subgroup of | #1833×D67, C2×#22560330, #30856006, … |
| maximal normal subgroup of | #1833×D67, C2×#22560330, #30856006, … |
| maximal quotient of | #30749474, #30749478, #30749482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4631);