Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #332×C67 ⋊ C2 |
| Sylow | C67 ⋊ #1699 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #14524016 |
| commutator | C42×C67 | C23 |
| Frattini | C42 | C22×D67 |
| Fitting | #332×C67 | C2 |
| socle | C22×C67 | #121 |
| hypercenter | #332 | D67 |
| hypocenter | C67 | #1699 |
| LCS | G ▷ C42×C67 ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #332 | |
160 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 49 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=dg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf,h^b=h^66,e^c=eg,h^c=h^66650 conjugacy classes — view the full character table.
| maximal subgroup of | #1699×D67, C2×#22559755, #30838412, … |
| maximal normal subgroup of | #1699×D67, C2×#22559755, #30838412, … |
| maximal quotient of | #30748934, #30752041, #30752051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4056);