Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62611 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3552 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6028 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)(5,12)(6,10)(7,16)(8,11)(9,14)(13,18)(15,17)(19,20,24,28)(21,25,30,38)(22,32,33,47)(23,35,36,27)(26,40,41,34)(29,42,43,50)(31,46,45,39)(37,48,49,44);(3,5)(4,9)(6,13)(7,15)(8,14)(10,17)(11,12)(16,18)(19,21,24,30)(20,25,28,38)(22,29,33,43)(23,31,36,45)(26,37,41,49)(27,39,35,46)(32,42,47,50)(34,44,40,48);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,12,18)(19,22,23,34,24,33,36,40)(20,26,27,32,28,41,35,47)(21,29,31,44,30,43,45,48)(25,37,39,42,38,49,46,50);(5,14)(9,12)(13,15)(17,18)(22,33)(26,41)(29,43)(32,47)(34,40)(37,49)(42,50)(44,48);(3,7,8,6)(4,10,11,16)(5,15,14,13)(9,17,12,18)(21,30)(22,33)(25,38)(26,41)(31,45)(32,47)(34,40)(39,46)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10487227);