Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C22×C42×C67 ⋊ C2 |
| Sylow | C67 ⋊ #1437 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C22×D67 |
| commutator | C22×C67 | C2×C42 |
| Fitting | C22×C42×C67 | C2 |
| socle | C24×C67 | C23 |
| hypercenter | C22×C42 | D67 |
| hypocenter | C67 | #1437 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C24 ◁ C22×C42 | |
828 conjugacy classes of subgroups; only the 241 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 241 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=g,c^2=ef,b^a=bd,c^a=ce,h^a=h^662168 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30747343, #1437×D67, C2×#22558447, … |
| maximal normal subgroup of | #30747343, #1437×D67, C2×#22558447, … |
| maximal quotient of | #30743751, #30743752, #30743754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 2748);