Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19504 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19528 |
| 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 | |
1560 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)(6,13)(9,14)(10,15)(12,17)(16,18)(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);(3,4,6,8,7,10,13,15)(5,9,12,14,11,16,17,18)(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)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(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);(5,11)(9,16)(12,17)(14,18)(21,30)(25,39)(29,44)(31,46)(38,49)(40,47)(43,50)(45,48)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261809);