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×#58476 |
| 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)(12,17)(19,20)(21,27)(22,29)(23,31)(24,25)(26,32)(28,30)(33,34)(35,36,41,43)(37,44,38,39)(40,48,49,42)(45,47,46,50);(4,8)(9,14)(10,15)(16,18)(20,25)(22,27)(23,32)(28,34)(35,37,41,38)(36,39,43,44)(40,45,49,46)(42,47,48,50);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,21,24,29)(20,22,25,27)(23,28,32,34)(26,30,31,33)(35,38,41,37)(36,39,43,44)(40,46,49,45)(42,47,48,50);(3,5,7,11)(4,9,8,14)(6,12,13,17)(10,16,15,18)(19,22)(20,21)(23,30)(24,27)(25,29)(26,28)(31,34)(32,33)(35,39)(36,37)(38,43)(40,47)(41,44)(42,45)(46,48)(49,50);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,23)(20,26)(21,28)(22,30)(24,32)(25,31)(27,33)(29,34)(35,40,41,49)(36,42,43,48)(37,45,38,46)(39,47,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011456);