Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61877 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5892 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)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,25,28)(21,26,31,42)(22,33,35,44)(23,29,24,37)(27,38,43,34)(30,45,46,50)(32,40,36,41)(39,47,49,48);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,26)(22,30)(23,36)(24,32)(25,31)(27,39)(28,42)(29,41)(33,45)(34,48)(35,46)(37,40)(38,47)(43,49)(44,50);(19,22)(20,27)(21,30)(23,34)(24,38)(25,35)(26,39)(28,43)(29,44)(31,46)(32,47)(33,37)(36,48)(40,45)(41,50)(42,49);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,28)(21,31)(27,43)(29,37)(30,46)(32,36)(33,44)(47,48);(4,10)(8,11)(9,18)(16,17)(19,23,25,24)(20,29,28,37)(21,32,31,36)(22,34,35,38)(26,40,42,41)(27,44,43,33)(30,47,46,48)(39,45,49,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485471);