Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C219 |
| direct product | C2×D4×#1438 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C24 | C25×C4 |
| Frattini, cosocle | C25 | C26 |
| prime radical | C24×D42 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
1030832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 22 points:
(4,5);(11,12)(13,14)(15,16,17,18);(1,2);(3,4,6,5)(8,10)(12,14)(19,20,21,22);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22);(3,5,6,4)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((4,5), (11,12)(13,14)(15,16,17,18), (1,2), (3,4,6,5)(8,10)(12,14)(19,20,21,22), (7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22), (3,5,6,4));