Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C23×C4×C127 ⋊ C2 |
| Sylow | C127 ⋊ #525 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D127 |
| commutator, socle | C22×C127 | C24 |
| Frattini | C22 | C23×D127 |
| Fitting | C23×C4×C127 | C2 |
| hypercenter | C23×C4 | D127 |
| hypocenter | C127 | #525 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 | |
330 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,127|d^2=e,b^a=be,c^a=cf,g^a=g^1262044 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30637219, #30637222, #30637232, … |
| maximal normal subgroup of | #30637219, #30637222, #30637232, … |
| maximal quotient of | #30641113, #30641173, #30641213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1060);