Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | #417×C127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×D4 |
| commutator | C4 | C22×C4×C127 |
| Frattini, cosocle | C2×C4 | C23×C127 |
| prime radical | C2×#41×C127 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ G | |
162 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2794 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C127×#1792, C127×#1793, C127×#1815, … |
| maximal normal subgroup of | C127×#1792, C127×#1793, C127×#1815, … |
| maximal quotient of | C127×#1477, C127×#1483, C127×#1489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 98);