Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213×C4 |
| direct product | C22×#3156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
12442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2);(3,4);(6,8)(9,10,14,18)(11,19,23,33)(12,16,24,30)(13,26,20,17)(15,27,29,22)(21,32,35,40)(25,37,34,31)(28,38,39,36);(9,11,20,27,14,23,13,22)(10,15,26,33,18,29,17,19)(12,21,34,38,24,35,25,36)(16,28,37,40,30,39,31,32);(5,6,7,8)(9,12)(10,16)(11,21)(13,25)(14,24)(15,28)(17,31)(18,30)(19,32)(20,34)(22,36)(23,35)(26,37)(27,38)(29,39)(33,40);(12,24)(16,30)(21,35)(25,34)(28,39)(31,37)(32,40)(36,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482435);