Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#339 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D4 |
| commutator | C4 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C23 | C22⋊C4 |
| prime radical | C24 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2);(3,4,5,6)(7,8)(9,11)(10,13)(12,14);(8,11,13,12)56 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#339, #10701, #10706, … |
| maximal normal subgroup of | C4×#339, #10701, #10706, … |
| maximal quotient of | C2×#1276, #7867, #7869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 464);