Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 20478689280 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C215 |
| abelian | C25×C4 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C26 |
| socle, prime radical | C26 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
5276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 14 points:
(1,2,3,4);(5,6);(7,8);(9,10);(11,12);(13,14)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2319);