Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18732 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2632 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);(5,11)(6,13)(8,16)(9,14)(12,15)(17,18)(19,20,25,30)(21,31,34,44)(22,27,36,42)(23,29,24,28)(26,38,40,33)(32,45,48,49)(35,43,37,41)(39,46,50,47);(3,4,7,10)(5,8,12,17)(6,14,13,9)(11,18,15,16)(19,21)(20,26)(22,32)(23,33)(24,38)(25,34)(27,39)(28,31)(29,44)(30,40)(35,46)(36,48)(37,47)(41,49)(42,50)(43,45);(3,5)(4,8)(6,15)(7,12)(9,18)(10,17)(11,13)(14,16)(19,22,25,36)(20,27,30,42)(21,32,34,48)(23,37,24,35)(26,39,40,50)(28,43,29,41)(31,45,44,49)(33,47,38,46);(19,23,25,24)(20,28,30,29)(21,33,34,38)(22,35,36,37)(26,31,40,44)(27,41,42,43)(32,46,48,47)(39,49,50,45)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255537);