Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C67 ⋊ #1791 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #14524012 |
| commutator | C23×C67 | C22×C4 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #585×C67 | C2 |
| socle | C2×C67 | #379 |
| hypercenter | #585 | D67 |
| hypocenter | C67 | #1791 |
| LCS | G ▷ C23×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #585 | |
548 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 121 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,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^66,e^b=eg,f^b=fg,d^c=dg1154 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30773121, #30773122, #30773128, … |
| maximal normal subgroup of | #30773121, #30773122, #30773128, … |
| maximal quotient of | #30747224, #30747227, #30747352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4461);