Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19572 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(4,8)(6,9)(10,14)(11,15)(13,16)(17,18)(19,20,24,28)(21,25,30,39)(22,32,34,42)(23,35,37,27)(26,36,41,33)(29,43,44,50)(31,47,46,40)(38,48,49,45);(3,4,9,15,7,11,6,8)(5,10,16,18,12,17,13,14)(19,21)(20,25)(22,29)(23,31)(24,30)(26,38)(27,40)(28,39)(32,43)(33,45)(34,44)(35,47)(36,48)(37,46)(41,49)(42,50);(3,5)(4,10)(6,13)(7,12)(8,14)(9,16)(11,17)(15,18)(19,22,24,34)(20,26,28,41)(21,29,30,44)(23,36,37,33)(25,38,39,49)(27,42,35,32)(31,48,46,45)(40,50,47,43);(4,11)(5,12)(8,15)(13,16)(21,30)(25,39)(29,44)(31,46)(38,49)(40,47)(43,50)(45,48)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261869);