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×#62225 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3565 | 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)(21,25)(22,32)(23,35)(24,27)(26,33)(28,37)(29,43)(30,39)(31,47)(34,42)(36,41)(38,45)(40,46)(44,50)(48,49);(3,5)(4,8)(6,13)(7,12)(9,16)(10,11)(14,17)(15,18)(19,21)(20,25)(22,29)(23,31)(24,30)(26,38)(27,39)(28,40)(32,43)(33,45)(34,44)(35,47)(36,48)(37,46)(41,49)(42,50);(19,22,24,34)(20,26,27,41)(21,29,30,44)(23,36,37,33)(25,38,39,49)(28,42,35,32)(31,48,46,45)(40,50,47,43);(5,12)(8,11)(13,18)(16,17)(20,27)(21,30)(26,41)(28,35)(29,44)(31,46)(32,42)(45,48);(3,6,7,15)(4,9,10,14)(5,13,12,18)(8,16,11,17)(21,30)(25,39)(29,44)(31,46)(38,49)(40,47)(43,50)(45,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486673);