Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #104×C127 ⋊ C2 |
| Sylow | C127 ⋊ #444 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D127 |
| commutator | C4×C127 | C22×C4 |
| Frattini | C2×C4 | C22×D127 |
| Fitting | #104×C127 | C2 |
| socle | C2×C127 | C4×D4 |
| hypercenter | #104 | D127 |
| hypocenter | C127 | #444 |
| LCS | G ▷ C4×C127 ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #104 | |
104 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,127|a^2=e,c^2=e,d^2=f,e^2=f,b^a=bd,c^a=cf,d^a=df,d^b=df,g^c=g^126904 conjugacy classes — view the full character table.
| maximal subgroup of | #444×D127, C2×#22549995, #30647628, … |
| maximal normal subgroup of | #444×D127, C2×#22549995, #30647628, … |
| maximal quotient of | #30637539, #30637545, #30637546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 668);