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×#19067 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2560 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);(3,4,8,5)(6,13,14,18)(7,12,17,10)(9,16,11,15)(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);(3,5,8,4)(6,11,14,9)(7,10,17,12)(13,16,18,15)(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);(3,6)(4,9)(5,11)(7,16)(8,14)(10,18)(12,13)(15,17)(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, 6258669);