Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C127⋊#338 |
| Sylow | C127 ⋊ #338 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×Dic127 |
| commutator | C4×C127 | C42 |
| Frattini | C22×C4 | C2×D127 |
| Fitting | C2×M16×C127 | C2 |
| socle | C2×C127 | #95 |
| hypercenter | C2×M16 | D127 |
| hypocenter | C127 | #338 |
| LCS | G ▷ C4×C127 ▷ C127 | |
| UCS | C1 ◁ C4 ◁ C2×M16 | |
86 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=d,b^2=e,c^2=f,d^2=f,e^2=f,b^a=bc,e^a=ef,d^b=df,g^b=g^1261282 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30636721, #30636722, #30636724, … |
| maximal normal subgroup of | #30636721, #30636722, #30636724, … |
| maximal quotient of | #30636560, #30636561, #30636601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 307);