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×#54570 |
| 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 | |
4164 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)(5,11)(6,8)(7,14)(9,12)(10,15)(13,16)(17,18)(19,20,23,24)(21,27,29,25)(22,30,31,26)(28,33,32,34)(35,36,41,39)(37,44,38,42)(40,47,48,43)(45,49,46,50);(4,6)(5,12)(10,14)(13,18)(20,24)(22,31)(25,27)(28,32)(35,37,41,38)(36,42,39,44)(40,45,48,46)(43,49,47,50);(3,5)(4,9)(6,11)(7,13)(8,12)(10,16)(14,17)(15,18)(19,21)(20,25)(22,32)(23,29)(24,27)(26,34)(28,31)(30,33)(35,38,41,37)(36,42,39,44)(40,46,48,45)(43,49,47,50);(3,6,8,4)(5,11,12,9)(7,14,15,10)(13,17,18,16)(19,22,23,31)(20,26,24,30)(21,28,29,32)(25,33,27,34)(35,39,41,36)(37,42,38,44)(40,47,48,43)(45,49,46,50);(3,7)(4,10)(5,13)(6,14)(8,15)(9,16)(11,17)(12,18)(19,20)(21,25)(22,26)(23,24)(27,29)(28,33)(30,31)(32,34)(35,40,41,48)(36,43,39,47)(37,45,38,46)(42,49,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010811);