Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C67 ⋊ #2082 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16863322 |
| commutator | C22×C4×C67 | C23 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #430×C67 | C2 |
| socle | C2×C67 | #397 |
| hypercenter | #430 | D67 |
| hypocenter | C67 | #2082 |
| LCS | G ▷ C22×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #430 | |
184 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 59 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=e,b^2=g,c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,h^b=h^66,e^c=eg,f^c=fg749 conjugacy classes — view the full character table.
| maximal subgroup of | #2082×D67, C2×#22561674, #30884780, … |
| maximal normal subgroup of | #2082×D67, C2×#22561674, #30884780, … |
| maximal quotient of | #30760625, #30760630, #30760642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 5975);