Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C22⋊C8×D127 |
| Fitting | C22⋊C8×C127 ⋊ C2 |
| Sylow | C127 ⋊ C2×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D127 |
| commutator | C2×C127 | C22×C8 |
| Fitting | C22⋊C8×C127 | C2 |
| socle | C22×C127 | C22×C4 |
| hypercenter | C22⋊C8 | D127 |
| hypocenter | C127 | C2×C22⋊C8 |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C2×C4 ◁ C22⋊C8 | |
202 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1300 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #323×D127, #324×D127, #30637045, … |
| maximal normal subgroup of | #323×D127, #324×D127, #30637045, … |
| maximal quotient of | #323×D127, #324×D127, #30637041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 508);