Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C222 |
| direct product | C22×#519568 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C23 | C26×C4 |
| Frattini, cosocle | C24 | C27 |
| prime radical | C23×#2408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1072392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 20 points:
(6,8)(9,10,11,12)(18,20);(17,18,19,20);(13,14,15,16)(17,18)(19,20);(3,4);(6,8)(10,12);(1,2)(3,4);(5,6)(7,8)(9,10)(11,12)560 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((6,8)(9,10,11,12)(18,20), (17,18,19,20), (13,14,15,16)(17,18)(19,20), (3,4), (6,8)(10,12), (1,2)(3,4), (5,6)(7,8)(9,10)(11,12));