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×#55849 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(4,8)(5,11)(10,15)(13,17)(19,20)(21,27)(22,29)(23,31)(24,25)(26,32)(28,30)(33,34)(35,36,40,43)(37,38,41,46)(39,42,48,50)(44,45,47,49);(6,14)(10,15)(13,17)(16,18)(20,25)(22,27)(23,32)(28,34)(35,37,40,41)(36,38,43,46)(39,47,48,44)(42,49,50,45);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21,24,29)(20,22,25,27)(23,28,32,34)(26,30,31,33)(35,37)(36,38)(39,44)(40,41)(42,45)(43,46)(47,48)(49,50);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,22)(20,21)(23,30)(24,27)(25,29)(26,28)(31,34)(32,33)(35,38)(36,41)(37,43)(39,45)(40,46)(42,47)(44,50)(48,49);(3,6)(4,10)(5,13)(7,14)(8,15)(9,16)(11,17)(12,18)(19,23,24,32)(20,26,25,31)(21,28,29,34)(22,30,27,33)(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, 8008880);