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×#56218 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4246 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)(5,9)(6,13)(7,8)(10,14)(11,17)(12,15)(16,18)(19,20,24,27)(21,22,29,25)(23,26,31,33)(28,30,34,32)(36,40)(37,43)(38,46)(48,50);(4,8)(9,15)(10,13)(16,17)(23,31)(26,33)(28,34)(30,32)(35,36,39,40)(37,44,43,41)(38,42,46,47)(45,50,49,48);(3,5,7,12)(4,9,8,15)(6,11,14,18)(10,16,13,17)(19,21,24,29)(20,25,27,22)(23,28,31,34)(26,32,33,30)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,4)(5,9)(6,10)(7,8)(11,16)(12,15)(13,14)(17,18)(19,22,24,25)(20,21,27,29)(23,30,31,32)(26,28,33,34)(35,37,39,43)(36,41,40,44)(38,45,46,49)(42,48,47,50);(3,6)(4,10)(5,11)(7,14)(8,13)(9,16)(12,18)(15,17)(19,23,24,31)(20,26,27,33)(21,28,29,34)(22,30,25,32)(35,38,39,46)(36,42,40,47)(37,45,43,49)(41,48,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010133);