Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22×C4 |
| direct product | #388×C127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C127 | C23 |
| commutator | C22 | C22×C4×C127 |
| Frattini, cosocle | C23 | C23×C127 |
| prime radical | C24×C127 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ G | |
190 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
3556 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#388×C127, C127×#2290, C127×#2293, … |
| maximal normal subgroup of | C2×#388×C127, C127×#2290, C127×#2293, … |
| maximal quotient of | C127×#1434, C127×#1436, C127×#1437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 69);