Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #135×C127 ⋊ C2 |
| Sylow | C127 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D127 |
| commutator | C4×C127 | C24 |
| Frattini | C4 | C23×D127 |
| Fitting | #135×C127 | C2 |
| socle | C2×C127 | C22×D4 |
| hypercenter | #135 | D127 |
| hypocenter | C127 | #575 |
| LCS | G ▷ C4×C127 ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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=f,e^2=f,b^a=be,c^a=cf,e^a=ef,e^b=ef,g^c=g^126904 conjugacy classes — view the full character table.
| maximal subgroup of | #30640674, #30640677, #30640679, … |
| maximal normal subgroup of | #30640674, #30640677, #30640679, … |
| maximal quotient of | #30647245, #30647318, #30647526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1346);