Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C228 |
| direct product | C22×#7631286 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C26 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5896744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 20 points:
(6,8)(10,12)(14,17)(15,19);(5,6)(7,8)(14,17)(15,19);(14,17)(15,19);(15,19)(18,20);(3,4);(1,2)(3,4);(9,10)(11,12)(13,14)(15,18)(16,17)(19,20);(5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)488 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)(10,12)(14,17)(15,19), (5,6)(7,8)(14,17)(15,19), (14,17)(15,19), (15,19)(18,20), (3,4), (1,2)(3,4), (9,10)(11,12)(13,14)(15,18)(16,17)(19,20), (5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20));