Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#16861188 |
| Sylow | C127 ⋊ C2×C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D127 |
| commutator | C22×C127 | C24 |
| Frattini | C22 | C23×D127 |
| Fitting | C23×C4×C127 | C2 |
| socle | C23×C127 | C23 |
| hypercenter | C23×C4 | D127 |
| hypocenter | C127 | C2×C22⋊Q8 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
322 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2044 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30637093, #30637095, C2×#22549612, … |
| maximal normal subgroup of | #30637093, #30637095, C2×#22549612, … |
| maximal quotient of | C2×#22549736, C2×#22549749, C2×#22549760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1047);