Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #421×C67 ⋊ C2 |
| Sylow | C67 ⋊ #1906 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #14524015 |
| commutator | C2×C4×C67 | C22×C4 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #421×C67 | C2 |
| socle | C2×C67 | #386 |
| hypercenter | #421 | D67 |
| hypocenter | C67 | #1906 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #421 | |
332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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,b^2=df,e^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eg,f^b=fg,h^b=h^66,d^c=dg,e^c=eg,f^c=fg758 conjugacy classes — view the full character table.
| maximal subgroup of | #1906×D67, C2×#22560708, #30857742, … |
| maximal normal subgroup of | #1906×D67, C2×#22560708, #30857742, … |
| maximal quotient of | #30751296, #30751303, #30751310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 5009);