Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 660602880 |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C24×D4 |
| unambiguous semidirect product | C4⋊C25 |
| unambiguous semidirect product | C22⋊C25 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C22 |
| commutator, Frattini, cosocle | C2 | C26 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
5276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 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, 2320);