Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19058 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2744 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);(3,4,7,10)(5,11,12,18)(6,14,15,9)(8,16,17,13)(19,20)(21,31)(22,36)(23,28)(24,29)(25,30)(26,34)(27,38)(32,40)(33,41)(35,44)(37,42)(39,43)(45,47)(46,49)(48,50);(6,15)(9,14)(11,18)(13,16)(19,21)(20,26)(22,32)(23,33)(24,34)(25,35)(27,40)(28,31)(29,44)(30,41)(36,49)(37,46)(38,47)(39,48)(42,50)(43,45);(3,5)(4,8)(6,16)(7,12)(9,18)(10,17)(11,14)(13,15)(19,22,25,39)(20,27,30,42)(21,32,35,48)(23,38,24,37)(26,40,41,50)(28,43,29,36)(31,45,44,49)(33,47,34,46);(19,23,25,24)(20,28,30,29)(21,33,35,34)(22,37,39,38)(26,31,41,44)(27,36,42,43)(32,46,48,47)(40,49,50,45)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258658);