Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C127 ⋊ #340 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | D508 |
| commutator | C4×C127 | C42 |
| Frattini | C22×C4 | C2×D127 |
| Fitting | C22×C8×C127 | C2 |
| socle | C23×C127 | D4 |
| hypercenter | C22×C8 | D127 |
| hypocenter | C127 | #340 |
| LCS | G ▷ C4×C127 ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×C8 | |
114 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=d,b^2=e,c^2=f,e^2=f,b^a=bc,c^a=cf,e^a=ef,g^a=g^1262044 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30636739, #30636740, #30636742, … |
| maximal normal subgroup of | #30636739, #30636740, #30636742, … |
| maximal quotient of | #30636565, #30636570, #30636589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 312);