Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #123×C127 ⋊ C2 |
| Sylow | C127 ⋊ #458 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16861177 |
| commutator | C23×C127 | C23 |
| Frattini | C23 | C22×D127 |
| Fitting | #123×C127 | C2 |
| socle | C2×C127 | #120 |
| hypercenter | #123 | D127 |
| hypocenter | C127 | #458 |
| LCS | G ▷ C23×C127 ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #123 | |
160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,127|a^2=f,b^a=bd,c^a=ce,e^b=ef,d^c=df,g^c=g^126898 conjugacy classes — view the full character table.
| maximal subgroup of | #30640720, #30640721, #30640726, … |
| maximal normal subgroup of | #30640720, #30640721, #30640726, … |
| maximal quotient of | #30637714, #30637715, #30637719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 738);