Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×D4×Dic127 |
| Sylow | C127 ⋊ C2×C4×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D127 |
| commutator | C2×C127 | C23×C4 |
| Frattini | C22 | C23×D127 |
| Fitting | C22×D4×C127 | C2 |
| socle | C23×C127 | C23 |
| hypercenter | C22×D4 | D127 |
| hypocenter | C127 | C2×C4×D4 |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
426 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 215 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 | #99×Dic127, #100×Dic127, #102×Dic127, … |
| maximal normal subgroup of | #99×Dic127, #100×Dic127, #102×Dic127, … |
| maximal quotient of | C2×C22⋊C4×Dic127, C2×C4⋊C4×Dic127, C4×D4×Dic127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1010);