Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72238724 (order 2048) · Inn(G) = #99 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #99 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| prime radical | C2×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2);(3,4)(5,8,10,7)(6,9);(4,7)(5,10)(8,9)26 conjugacy classes — view the full character table.
| maximal subgroup of | #7485, #7493, #7498, … |
| maximal normal subgroup of | #7485, #7493, #7498, … |
| maximal quotient of | C2×#1318, C2×#1325, #11475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 856);