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×#19060 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2674 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);(5,11)(6,13)(8,16)(9,14)(12,15)(17,18)(19,20,25,30)(21,31,34,43)(22,35,36,44)(23,29,24,28)(26,38,41,33)(27,39,42,37)(32,40,46,50)(45,47,49,48);(3,4,7,10)(5,8,12,17)(6,14,13,9)(11,18,15,16)(19,21)(20,26)(22,32)(23,38)(24,33)(25,34)(27,40)(28,43)(29,31)(30,41)(35,49)(36,46)(37,48)(39,47)(42,50)(44,45);(3,5)(4,8)(6,15)(7,12)(9,18)(10,17)(11,13)(14,16)(19,22)(20,27)(21,32)(23,37)(24,39)(25,36)(26,40)(28,35)(29,44)(30,42)(31,45)(33,47)(34,46)(38,48)(41,50)(43,49);(22,36)(27,42)(32,46)(35,44)(37,39)(40,50)(45,49)(47,48)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258661);