Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×C4×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22 |
| commutator | C2 | C24×C4 |
| Frattini, cosocle | C22 | C25 |
| socle | C24 | C23 |
| prime radical | C23×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 12 points:
(1,2);(3,4);(5,6,7,8);(9,10,11,12);(10,12)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2154);