Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C222 |
| direct product | C22×D4×#394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator, Frattini, cosocle | C24 | C27 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1098792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 20 points:
(18,19);(17,18,20,19);(10,12)(14,16)(18,19);(6,8)(10,12)(13,14)(15,16)(17,19,20,18);(3,4);(5,6,7,8)(9,10)(11,12);(1,2)(3,4)440 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((18,19), (17,18,20,19), (10,12)(14,16)(18,19), (6,8)(10,12)(13,14)(15,16)(17,19,20,18), (3,4), (5,6,7,8)(9,10)(11,12), (1,2)(3,4));