Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
| direct product | C4×#1979 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#121 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4);(6,8)(9,10,14,19,15,18,13,27)(11,16,22,31,21,32,28,30)(12,23,25,33,26,37,24,17)(20,34,36,29,35,39,38,40);(5,6,7,8)(9,11,13,22,15,21,14,28)(10,16,19,30,18,32,27,31)(12,20,24,36,26,35,25,38)(17,29,23,34,33,40,37,39);(9,12,14,25,15,26,13,24)(10,17,27,37,18,33,19,23)(11,20,28,38,21,35,22,36)(16,29,31,39,32,40,30,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261796);