Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19732 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C24 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1856 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)(5,13)(6,9)(7,17)(8,12)(10,15)(11,14)(16,18)(19,20,24,28)(21,29,33,43)(22,26,34,40)(23,36,30,27)(25,37,39,32)(31,42,45,50)(35,47,44,41)(38,48,49,46);(3,5,14,16,8,15,7,6)(4,9,17,10,12,18,11,13)(19,21,30,37,24,33,23,32)(20,25,36,43,28,39,27,29)(22,31,44,48,34,45,35,46)(26,38,47,50,40,49,41,42);(3,6,7,15,8,16,14,5)(4,10,11,9,12,13,17,18)(19,22,24,34)(20,26,28,40)(21,31,33,45)(23,35,30,44)(25,38,39,49)(27,41,36,47)(29,42,43,50)(32,46,37,48);(5,15)(6,16)(9,18)(10,13)(22,34)(26,40)(31,45)(35,44)(38,49)(41,47)(42,50)(46,48)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264566);