Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19091 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4)(5,11)(6,14)(7,9)(8,13)(10,15)(12,18)(16,17)(19,20,25,30)(21,31,34,41)(22,35,37,43)(23,28,24,29)(26,38,42,33)(27,39,44,36)(32,40,48,50)(45,46,49,47);(6,15)(10,14)(11,18)(13,16)(19,21,25,34)(20,26,30,42)(22,32,37,48)(23,38,24,33)(27,40,44,50)(28,31,29,41)(35,49,43,45)(36,47,39,46);(3,5)(4,8)(6,16)(7,12)(9,17)(10,18)(11,14)(13,15)(19,22)(20,27)(21,32)(23,36)(24,39)(25,37)(26,40)(28,43)(29,35)(30,44)(31,45)(33,46)(34,48)(38,47)(41,49)(42,50);(4,9)(8,17)(10,14)(11,18)(19,23,25,24)(20,28,30,29)(21,33,34,38)(22,36,37,39)(26,41,42,31)(27,43,44,35)(32,46,48,47)(40,49,50,45)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258905);