Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C26 |
| direct product | C2×#72224639 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #12067 |
| commutator | C16 | C23×C16 |
| Frattini, cosocle | #1311 | C24 |
| prime radical | C22×D32 | C8 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ C2×#7053 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,6,10,7,11,13,17,8,12,16,5,9,14,18,15)(20,25,27,36,28,37,40,39)(21,29,31,44)(22,32)(23,34)(26,35,41,38,42,48,50,49)(30,43,46,45)(33,47);(3,5,13,4,9,17,6,14,8,10,18,12,7,15,16,11)(20,26,40,50,28,42,27,41)(21,30,31,46)(22,33)(25,35,39,49,37,48,36,38)(29,43,44,45)(32,47);(19,20,21,25,22,26,29,38,23,27,30,39,32,41,45,35,24,28,31,37,33,42,44,49,34,40,46,36,47,50,43,48)188 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((1,2), (3,4,6,10,7,11,13,17,8,12,16,5,9,14,18,15)(20,25,27,36,28,37,40,39)(21,29,31,44)(22,32)(23,34)(26,35,41,38,42,48,50,49)(30,43,46,45)(33,47), (3,5,13,4,9,17,6,14,8,10,18,12,7,15,16,11)(20,26,40,50,28,42,27,41)(21,30,31,46)(22,33)(25,35,39,49,37,48,36,38)(29,43,44,45)(32,47), (19,20,21,25,22,26,29,38,23,27,30,39,32,41,45,35,24,28,31,37,33,42,44,49,34,40,46,36,47,50,43,48));