Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #126×C127 ⋊ C2 |
| Sylow | C127 ⋊ #560 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D127 |
| commutator, socle | C22×C127 | C24 |
| Fitting | #126×C127 | C2 |
| hypercenter | #126 | D127 |
| hypocenter | C127 | #560 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ #126 | |
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,127|b^a=be,c^a=cf,d^a=def,g^a=g^126,c^b=cef,d^b=de,g^b=g^126,g^c=g^126901 conjugacy classes — view the full character table.
| maximal subgroup of | #560×D127, C2×#22550617, #30652869, … |
| maximal normal subgroup of | #560×D127, C2×#22550617, #30652869, … |
| maximal quotient of | #30641870, #30641877, #30641883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1290);