Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(5,11)(6,13)(8,16)(9,14)(12,15)(17,18)(19,20,25,30)(21,31,34,42)(22,27,38,45)(23,29,24,28)(26,35,41,33)(32,47,48,43)(36,50,49,40)(37,46,39,44);(3,4,7,10)(5,8,12,17)(6,14,13,9)(11,18,15,16)(19,21)(20,26)(22,36)(23,35)(24,33)(25,34)(27,43)(28,42)(29,31)(30,41)(32,37)(38,49)(39,48)(40,44)(45,47)(46,50);(3,5)(4,8)(6,15)(7,12)(9,18)(10,17)(11,13)(14,16)(19,22,25,38)(20,27,30,45)(21,32,34,48)(23,37,24,39)(26,40,41,50)(28,44,29,46)(31,43,42,47)(33,36,35,49)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143916);