Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#9757 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4)(5,11)(6,14)(7,9)(8,12)(10,15)(13,17)(16,18)(19,20)(21,25,30,39)(22,32)(23,36)(24,27)(26,34)(28,33)(29,41,43,50)(31,45,42,38)(35,46)(37,48,49,44)(40,47);(3,5,6,12,7,13,15,18)(4,8,10,11,9,16,14,17)(19,21)(20,25,27,39)(22,29)(23,31)(24,30)(26,37,40,49)(28,38,36,45)(32,41,46,50)(33,42)(34,44)(35,43)(47,48);(4,9)(8,16)(10,14)(11,17)(19,22,33,47,24,35,23,34)(20,26,36,46,27,40,28,32)(21,29,42,48,30,43,31,44)(25,37,45,50,39,49,38,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112730);