Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | #123×D127 |
| Fitting | #123×C127 ⋊ C2 |
| Sylow | C127 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D127 |
| commutator, socle | C22×C127 | C24 |
| Fitting | #123×C127 | C2 |
| hypercenter | #123 | D127 |
| hypocenter | C127 | C2×#123 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ #123 | |
342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
910 conjugacy classes — view the full character table.
| maximal subgroup of | #352×D127, #458×D127, C2×#123×D127, … |
| maximal normal subgroup of | #352×D127, #458×D127, C2×#123×D127, … |
| maximal quotient of | #381×D127, #385×D127, #386×D127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1049);