Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C64 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C127⋊C64 |
| derived, Sylow | C127 ⋊ C64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | D127 |
| commutator, hypocenter | C127 | C64 |
| Fitting | C32×C127 | C2 |
| socle | C2×C127 | C32 |
| LCS | G ▷ C127 | |
| UCS | C1 ◁ C32 | |
14 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^a=g^1262080 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C64×D127, C2×C127⋊C64, #30636940, … |
| maximal normal subgroup of | C64×D127, C2×C127⋊C64, #30636940, … |
| maximal quotient of | C127⋊C128, C2×C127⋊C64 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 268);