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×#62234 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3516 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5990 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);(3,4,7,10)(5,11,12,8)(6,14,15,9)(13,16,18,17)(19,20)(21,29)(22,33)(23,27)(24,37)(25,31)(26,35)(28,36)(30,45)(32,40)(34,42)(38,41)(39,49)(43,47)(44,48)(46,50);(3,5)(4,8)(6,13)(7,12)(9,16)(10,11)(14,17)(15,18)(19,21)(20,25)(22,30)(23,31)(24,32)(26,39)(27,29)(28,40)(33,50)(34,46)(35,47)(36,48)(37,44)(38,49)(41,43)(42,45);(19,22,23,34)(20,26,27,41)(21,30,31,46)(24,38,36,35)(25,39,29,43)(28,42,37,33)(32,49,48,47)(40,45,44,50);(5,12)(8,11)(13,18)(16,17)(20,27)(25,29)(26,41)(28,37)(33,42)(39,43)(40,44)(45,50);(3,6,7,15)(4,9,10,14)(5,13,12,18)(8,16,11,17)(20,27)(21,31)(26,41)(28,37)(30,46)(32,48)(33,42)(47,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486683);