Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C127 ⋊ #346 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C2×D127 |
| commutator | C2×C127 | C4×C8 |
| Fitting | C2×C16×C127 | C2 |
| socle | C22×C127 | C2×C8 |
| hypercenter | C2×C16 | D127 |
| hypocenter | C127 | #346 |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C16 | |
54 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=c,b^2=d,c^2=e,e^2=f,b^a=bf,g^b=g^1262056 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30636865, #346×D127, C2×#22549651, … |
| maximal normal subgroup of | #30636865, #346×D127, C2×#22549651, … |
| maximal quotient of | #30636646, #30636649, #30636807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 324);