Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C127 · Schur multiplier (H2) = C23 |
| direct product | #340×C127 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C127 | D4 |
| commutator | C4 | C42×C127 |
| Frattini, cosocle | C22×C4 | C22×C127 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C22×C4×C127 ◁ G | |
114 conjugacy classes of subgroups.
3556 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C127×#1347, C127×#1348, C2×#340×C127, … |
| maximal normal subgroup of | C127×#1347, C127×#1348, C2×#340×C127, … |
| maximal quotient of | C127×#1277, C127×#1279, C127×#1286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 21);