Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4360 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);(5,11)(9,14)(12,17)(16,18)(20,24)(21,27)(26,31)(28,33)(35,36)(37,43)(38,40)(39,46)(41,44)(42,47)(45,48)(49,50);(4,8)(6,13)(9,14)(12,17)(20,24)(25,30)(26,31)(32,34)(36,38)(37,44)(42,46)(45,50);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,20,23,24)(21,25,27,30)(22,26,29,31)(28,32,33,34)(35,37)(36,41)(38,43)(39,45)(40,44)(42,48)(46,49)(47,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,21,23,27)(20,25,24,30)(22,28,29,33)(26,32,31,34)(35,38,40,36)(37,43,44,41)(39,46,47,42)(45,49,50,48);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,22,23,29)(20,26,24,31)(21,28,27,33)(25,32,30,34)(35,39,40,47)(36,42,38,46)(37,45,44,50)(41,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013137);