Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #335×C67 ⋊ C2 |
| Sylow | C67 ⋊ #1565 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4×D67 |
| commutator | C4×C67 | C22×C8 |
| Frattini | C42 | C22×D67 |
| Fitting | #335×C67 | C2 |
| socle | C22×C67 | C2×C4⋊C4 |
| hypercenter | #335 | D67 |
| hypocenter | C67 | #1565 |
| LCS | G ▷ C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ #335 | |
180 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 83 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=d,d^2=f,e^2=g,b^a=bd,d^a=df,c^b=cf,h^b=h^66,h^c=h^66968 conjugacy classes — view the full character table.
| maximal subgroup of | #1565×D67, C2×#22558960, #30824812, … |
| maximal normal subgroup of | #1565×D67, C2×#22558960, #30824812, … |
| maximal quotient of | #30745291, #30747027, #30747030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 3261);