Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | #405×C67 ⋊ C2 |
| Sylow | C67 ⋊ #3156 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D268 |
| commutator | C2×C4×C67 | C24 |
| Frattini | C2×C4 | C23×D67 |
| Fitting | #405×C67 | C2 |
| socle | C22×C67 | C22×D4 |
| hypercenter | #405 | D67 |
| hypocenter | C67 | #3156 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 | |
532 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 165 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=g,b^2=eg,c^2=f,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^66,d^c=dg1352 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30774106, #30774108, #30774720, … |
| maximal normal subgroup of | #30774106, #30774108, #30774720, … |
| maximal quotient of | #30843707, #30843751, #30843769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 15004);