Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C67 ⋊ #2873 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D67 |
| commutator | C2×C67 | C22×C42 |
| Fitting | C2×#117×C67 | C2 |
| socle | C22×C67 | C23×C4 |
| hypercenter | C2×#117 | D67 |
| hypocenter | C67 | #2873 |
| LCS | G ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 | |
564 conjugacy classes of subgroups; only the 373 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 373 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=f,e^2=g,d^a=dg,h^a=h^66,c^b=cg1388 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30753164, #30753167, #30753169, … |
| maximal normal subgroup of | #30753164, #30753167, #30753169, … |
| maximal quotient of | #30821415, #30821439, #30821468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 12929);