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×#57742 |
| 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 | |
4164 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);(4,8)(5,11)(10,15)(13,17)(19,20,24,21)(22,25,29,27)(23,28,31,26)(30,33,34,32)(35,36,41,38)(37,39,42,45)(40,43,49,46)(44,47,48,50);(6,14)(10,15)(13,17)(16,18)(20,21)(22,29)(23,31)(32,33)(35,37,41,42)(36,39,38,45)(40,48,49,44)(43,50,46,47);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21,24,20)(22,25,29,27)(23,28,31,26)(30,32,34,33)(35,38,41,36)(37,39,42,45)(40,46,49,43)(44,47,48,50);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,17)(15,18)(19,22,24,29)(20,25,21,27)(23,30,31,34)(26,32,28,33)(35,39)(36,42)(37,38)(40,47)(41,45)(43,48)(44,46)(49,50);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,40)(36,43)(37,44)(38,46)(39,47)(41,49)(42,48)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008943);