Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C127 ⋊ #385 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D127 |
| commutator | C22×C127 | C22×C4 |
| Fitting | C23×C4×C127 | C2 |
| socle | C23×C127 | C23 |
| hypercenter | C23×C4 | D127 |
| hypocenter | C127 | #385 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
234 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 87 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=f,b^2=f,b^a=bd,c^a=ce,g^a=g^1262044 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30639009, #30639010, #30639979, … |
| maximal normal subgroup of | #30639009, #30639010, #30639979, … |
| maximal quotient of | #30636953, #30636963, #30636973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 422);