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×#57063 |
| 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 | |
4116 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)(21,23,26,32)(22,25,31,34)(28,30,29,33)(36,37)(38,45)(39,47)(48,49);(4,8)(9,15)(10,13)(16,17)(21,26)(23,32)(28,29)(30,33)(35,36,40,37)(38,43,45,41)(39,44,47,42)(46,48,50,49);(3,5,7,12)(4,9,8,15)(6,11,14,18)(10,16,13,17)(19,21,24,26)(20,23,27,32)(22,29,31,28)(25,33,34,30)(35,37,40,36)(38,41,45,43)(39,44,47,42)(46,48,50,49);(3,4)(5,9)(6,10)(7,8)(11,16)(12,15)(13,14)(17,18)(19,22)(20,25)(21,28)(23,30)(24,31)(26,29)(27,34)(32,33)(35,38,40,45)(36,41,37,43)(39,46,47,50)(42,48,44,49);(3,6,7,14)(4,10,8,13)(5,11,12,18)(9,16,15,17)(19,23)(20,26)(21,27)(22,30)(24,32)(25,29)(28,34)(31,33)(35,39)(36,42)(37,44)(38,46)(40,47)(41,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006945);