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×#19097 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3567 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2694 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);(5,12)(6,7)(9,16)(10,14)(13,15)(17,18)(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);(3,4)(5,9)(6,10)(7,14)(8,11)(12,17)(13,18)(15,16)(19,21,25,33)(20,26,30,41)(22,32,37,46)(23,38,24,34)(27,40,42,50)(28,43,29,31)(35,49,44,45)(36,48,39,47);(3,5)(4,9)(6,12)(7,15)(8,13)(10,17)(11,18)(14,16)(19,22)(20,27)(21,32)(23,36)(24,39)(25,37)(26,40)(28,35)(29,44)(30,42)(31,45)(33,46)(34,47)(38,48)(41,50)(43,49);(3,6,8,7)(4,10,11,14)(5,12,13,15)(9,17,18,16)(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, 6258913);