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×#58567 |
| 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 | |
4148 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);(3,4,7,8)(5,11,12,9)(6,14,15,10)(13,16,18,17)(19,20)(21,25)(22,30)(23,24)(26,31)(27,28)(29,34)(32,33)(35,36,41,39)(37,44,38,42)(40,47,48,43)(45,49,46,50);(4,8)(5,12)(10,14)(13,18)(20,24)(21,27)(22,31)(32,34)(35,37,41,38)(36,42,39,44)(40,45,48,46)(43,49,47,50);(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(17,18)(19,21,23,27)(20,25,24,28)(22,29,31,33)(26,32,30,34)(35,38,41,37)(36,42,39,44)(40,46,48,45)(43,49,47,50);(3,5,7,12)(4,9,8,11)(6,13,15,18)(10,16,14,17)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33)(35,39,41,36)(37,42,38,44)(40,47,48,43)(45,49,46,50);(3,6,7,15)(4,10,8,14)(5,13,12,18)(9,16,11,17)(19,22)(20,26)(21,29)(23,31)(24,30)(25,32)(27,33)(28,34)(35,40)(36,43)(37,45)(38,46)(39,47)(41,48)(42,49)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011506);