Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C8 |
No aliases for this group.
| Sylow | C67 ⋊ #1715 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #14524039 |
| commutator | C42×C67 | C23 |
| Frattini | C42 | C22×D67 |
| Fitting | #493×C67 | C2 |
| socle | C22×C67 | #127 |
| hypercenter | #493 | D67 |
| hypocenter | C67 | #1715 |
| LCS | G ▷ C42×C67 ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|b^2=df,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf,h^c=h^66752 conjugacy classes — view the full character table.
| maximal subgroup of | #30743696, #30743698, #30769600, … |
| maximal normal subgroup of | #30743696, #30743698, #30769600, … |
| maximal quotient of | #30749617, #30755607, #30755621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4136);