Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D127)⋊C16 |
| Fitting | C2×C16×C127 ⋊ C2 |
| Sylow | C127 ⋊ C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C2×D127 |
| commutator | C2×C127 | C2×C16 |
| Fitting | C2×C16×C127 | C2 |
| socle | C22×C127 | C2×C8 |
| hypercenter | C2×C16 | D127 |
| hypocenter | C127 | C22⋊C16 |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C16 | |
66 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=d,d^2=e,e^2=f,b^a=bc,g^b=g^1262056 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22⋊C16×D127, C2×(C2×D127)⋊C16, #30640339, … |
| maximal normal subgroup of | C22⋊C16×D127, C2×(C2×D127)⋊C16, #30640339, … |
| maximal quotient of | #30636653, #30636669, #30636684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 328);