Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#16861255 |
| Fitting | C2×#41×C127 ⋊ C2 |
| Sylow | C127 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D127 |
| commutator | C2×C127 | C25 |
| Frattini | C2 | C24×D127 |
| Fitting | C2×#41×C127 | C2 |
| socle | C22×C127 | C24 |
| hypercenter | C2×#41 | D127 |
| hypocenter | C127 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1294 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30638682, #30638685, #30638698, … |
| maximal normal subgroup of | #30638682, #30638685, #30638698, … |
| maximal quotient of | C2×#22550348, C2×#22550350, C2×#22550356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1380);