Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| prime radical | C26 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(4,6)(7,8,9,10)(12,14);(11,12)(13,14);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1009);