Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊(C127⋊C16) |
| Sylow | C127 ⋊ C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C2×D127 |
| commutator | C2×C127 | C2×C16 |
| Fitting | C22×C8×C127 | C2 |
| socle | C22×C127 | C2×C8 |
| hypercenter | C22×C8 | D127 |
| hypocenter | C127 | C22⋊C16 |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 | |
66 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=d,d^2=e,e^2=f,b^a=bc,g^a=g^1262056 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30636653, #30636654, #30636655, … |
| maximal normal subgroup of | #30636653, #30636654, #30636655, … |
| maximal quotient of | #30636654, #30636670, D4⋊(C127⋊C16), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 329);