Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3559 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4)(5,11)(6,14)(7,9)(8,13)(10,15)(12,18)(16,17)(19,20,25,30)(21,31,33,43)(22,35,37,44)(23,29,24,28)(26,38,41,34)(27,39,42,36)(32,40,46,50)(45,47,49,48);(6,15)(10,14)(11,18)(13,16)(19,21)(20,26)(22,32)(23,38)(24,34)(25,33)(27,40)(28,43)(29,31)(30,41)(35,49)(36,48)(37,46)(39,47)(42,50)(44,45);(3,5)(4,8)(6,16)(7,12)(9,17)(10,18)(11,14)(13,15)(19,22,25,37)(20,27,30,42)(21,32,33,46)(23,36,24,39)(26,40,41,50)(28,35,29,44)(31,45,43,49)(34,47,38,48);(4,9)(8,17)(10,14)(11,18)(21,33)(26,41)(31,43)(32,46)(34,38)(40,50)(45,49)(47,48)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258903);