Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C22×#99 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22⋊C4 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| prime radical | C23×D4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2);(3,4);(5,6)(7,11,12,9)(8,10);(5,7)(6,9)(8,12)(10,11)44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1613);