Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56226 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4,8,5)(6,9,13,11)(7,12,15,10)(14,17,18,16)(19,20,24,27)(21,22,29,25)(23,26,31,33)(28,30,34,32)(35,36,40,43)(37,38,41,46)(39,42,48,50)(44,45,47,49);(4,5)(6,13)(7,15)(16,17)(23,31)(26,33)(28,34)(30,32)(35,37,40,41)(36,38,43,46)(39,47,48,44)(42,49,50,45);(3,5,8,4)(6,9,13,11)(7,12,15,10)(14,16,18,17)(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30)(35,37)(36,38)(39,44)(40,41)(42,45)(43,46)(47,48)(49,50);(3,6,8,13)(4,9,5,11)(7,14,15,18)(10,16,12,17)(19,22,24,25)(20,21,27,29)(23,30,31,32)(26,28,33,34)(35,38)(36,41)(37,43)(39,45)(40,46)(42,47)(44,50)(48,49);(3,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,17)(13,18)(19,23,24,31)(20,26,27,33)(21,28,29,34)(22,30,25,32)(35,39)(36,42)(37,44)(38,45)(40,48)(41,47)(43,50)(46,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010341);