Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C127 ⋊ #336 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C2×D127 |
| commutator | C2×C127 | C4×C8 |
| Fitting | C22×C8×C127 | C2 |
| socle | C23×C127 | C2×C4 |
| hypercenter | C22×C8 | D127 |
| hypocenter | C127 | #336 |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C8 | |
118 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=d,b^2=e,d^2=f,b^a=bc,g^b=g^1262056 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #336×D127, C2×#22549630, #30638422, … |
| maximal normal subgroup of | #336×D127, C2×#22549630, #30638422, … |
| maximal quotient of | C127⋊#1273, #30636558, #30636560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 303);