Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#72320485 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×#442 |
| commutator | C22×C8 | C24×C4 |
| Frattini, cosocle | C2×C4×C8 | C25 |
| prime radical | C22×#466 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ C22×#1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
4962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,16,20)(13,17,22,25)(14,23,21,18)(15,19,24,26)(28,32)(33,38)(34,39)(40,41);(3,5)(4,7)(6,10)(8,9)(11,13,15,21,16,22,24,14)(12,17,19,18,20,25,26,23)(28,33,32,38)(29,35)(30,37)(34,40,39,41)(36,42);(5,10)(7,9)(11,14,24,22,16,21,15,13)(12,18,26,17,20,23,19,25)(28,34,32,39)(29,36)(33,40,38,41)(35,42);(4,8)(7,9)(27,28,29,33,30,34,35,41,31,32,36,38,37,39,42,40)176 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)(5,9)(6,8)(7,10)(11,12,16,20)(13,17,22,25)(14,23,21,18)(15,19,24,26)(28,32)(33,38)(34,39)(40,41), (3,5)(4,7)(6,10)(8,9)(11,13,15,21,16,22,24,14)(12,17,19,18,20,25,26,23)(28,33,32,38)(29,35)(30,37)(34,40,39,41)(36,42), (5,10)(7,9)(11,14,24,22,16,21,15,13)(12,18,26,17,20,23,19,25)(28,34,32,39)(29,36)(33,40,38,41)(35,42), (4,8)(7,9)(27,28,29,33,30,34,35,41,31,32,36,38,37,39,42,40));