Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C4×C16×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2162 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D67 |
| commutator | C2×C67 | C22×C16 |
| Frattini | C2×C8 | C22×D67 |
| Fitting | C4×C16×C67 | C2 |
| socle | C22×C67 | C22×C8 |
| hypercenter | C4×C16 | D67 |
| hypocenter | C67 | #2162 |
| LCS | G ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C4×C8 ◁ C4×C16 | |
160 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 160 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=e,c^2=d,e^2=f,f^2=g,b^a=bd,h^b=h^662192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C16×#11761611, #30764844, #30764847, … |
| maximal normal subgroup of | C16×#11761611, #30764844, #30764847, … |
| maximal quotient of | #30770829, C4×(C2×D67)⋊C16, C4×Dic67⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 6307);