Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C220 |
| direct product | C22×#59051 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
128861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 20 points:
(10,11)(13,14)(15,17)(16,18)(19,20);(5,6)(7,8)(10,11)(14,16)(15,19);(6,8)(9,10,12,11)(13,15)(14,17)(16,20)(18,19);(5,6)(7,8)(9,11,12,10)(13,16,18,14)(15,20,19,17);(13,17,18,20)(14,19,16,15);(3,4);(1,2)(3,4)256 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((10,11)(13,14)(15,17)(16,18)(19,20), (5,6)(7,8)(10,11)(14,16)(15,19), (6,8)(9,10,12,11)(13,15)(14,17)(16,20)(18,19), (5,6)(7,8)(9,11,12,10)(13,16,18,14)(15,20,19,17), (13,17,18,20)(14,19,16,15), (3,4), (1,2)(3,4));