Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C210 · Schur multiplier (H2) = C245 |
| direct product | C28×D4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C29 | C22 |
| commutator, Frattini, cosocle | C2 | C210 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C29 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
451227752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 10 generators on 20 points:
(18,20);(13,14)(15,16);(17,18,19,20);(13,15)(14,16);(9,10)(11,12);(9,11)(10,12);(5,6)(7,8);(5,7)(6,8);(1,2)(3,4);(1,3)(2,4)1280 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((18,20), (13,14)(15,16), (17,18,19,20), (13,15)(14,16), (9,10)(11,12), (9,11)(10,12), (5,6)(7,8), (5,7)(6,8), (1,2)(3,4), (1,3)(2,4));