Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #344×C67 ⋊ C2 |
| Sylow | C67 ⋊ #1915 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16863274 |
| commutator | C23×C67 | C22×C4 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #344×C67 | C2 |
| socle | C2×C67 | #386 |
| hypercenter | #344 | D67 |
| hypocenter | C67 | #1915 |
| LCS | G ▷ C23×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #344 | |
308 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 79 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=dfg,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=eg,f^b=fg,h^b=h^66,d^c=dg,f^c=fg,h^c=h^66755 conjugacy classes — view the full character table.
| maximal subgroup of | #1915×D67, C2×#22560760, #30857862, … |
| maximal normal subgroup of | #1915×D67, C2×#22560760, #30857862, … |
| maximal quotient of | #30751370, #30751380, #30751823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 5061);