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×#61798 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3512 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6126 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)(19,20,25,30)(21,26,34,44)(22,28,36,46)(23,29,24,39)(27,38,47,35)(31,48,32,41)(33,42,37,43)(40,49,45,50);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21)(20,26)(22,32)(23,37)(24,33)(25,34)(27,45)(28,41)(29,43)(30,44)(31,36)(35,49)(38,50)(39,42)(40,47)(46,48);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22)(20,27)(21,31)(23,35)(24,38)(25,36)(26,40)(28,39)(29,46)(30,47)(32,34)(33,49)(37,50)(41,43)(42,48)(44,45);(5,13)(8,11)(12,18)(15,17)(19,22)(20,28)(21,32)(23,38)(24,35)(25,36)(26,41)(27,39)(29,47)(30,46)(31,34)(33,49)(37,50)(40,43)(42,45)(44,48);(4,10)(8,11)(9,16)(15,17)(19,23,25,24)(20,29,30,39)(21,33,34,37)(22,35,36,38)(26,42,44,43)(27,46,47,28)(31,49,32,50)(40,48,45,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485170);