Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×C8×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22 |
| commutator | C2 | C23×C8 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C23 | C22×C4 |
| prime radical | C22×D4 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4,5,6,7,8,9,10);(11,12,13,14);(12,14)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1658);