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×#57692 |
| 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 | |
4034 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);(3,4)(5,9)(6,14)(7,8)(10,15)(11,12)(13,18)(16,17)(19,20,24,27)(21,22,29,25)(23,26,31,33)(28,30,34,32)(35,36,41,43)(37,44,38,39)(40,48,49,42)(45,47,46,50);(4,8)(5,11)(6,15)(16,18)(23,31)(26,33)(28,34)(30,32)(35,37,41,38)(36,39,43,44)(40,45,49,46)(42,47,48,50);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30)(35,38,41,37)(36,39,43,44)(40,46,49,45)(42,47,48,50);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,18)(15,17)(19,22,24,25)(20,21,27,29)(23,30,31,32)(26,28,33,34)(35,39)(36,37)(38,43)(40,47)(41,44)(42,45)(46,48)(49,50);(3,6,7,15)(4,10,8,14)(5,13,11,17)(9,16,12,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,40,41,49)(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, 8008935);