Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #493×C67 ⋊ C2 |
| Sylow | C67 ⋊ D4×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D67 |
| commutator | C2×C4×C67 | C24 |
| Frattini | C2×C4 | C23×D67 |
| Fitting | #493×C67 | C2 |
| socle | C22×C67 | C22×D4 |
| hypercenter | #493 | D67 |
| hypocenter | C67 | D4×D8 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,d^b=de,e^b=eg,h^b=h^66,h^c=h^66,e^d=eg,h^d=h^66761 conjugacy classes — view the full character table.
| maximal subgroup of | #30769252, #30769256, #30769280, … |
| maximal normal subgroup of | #30769252, #30769256, #30769280, … |
| maximal quotient of | #30845177, #30845192, #30845207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 16185);