Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#22277 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 46 points:
(1,2);(3,4,6,5)(7,8)(9,12)(10,13)(11,14)(15,16)(17,25)(18,29)(19,23)(20,34)(21,27)(22,32)(24,33)(26,40)(28,36)(30,44)(31,43)(35,45)(37,41)(38,46)(39,42);(3,5,6,4)(7,9,12,8)(10,13,11,14)(15,17)(16,21)(18,30)(19,27)(20,28)(22,37)(23,25)(24,36)(26,31)(29,40)(32,45)(33,42)(34,39)(35,38)(41,46)(43,44);(7,10,12,11)(8,13,9,14)(15,18,20,32,19,31,33,46)(16,22,24,29,23,38,34,43)(17,26,28,41,27,30,42,45)(21,35,36,44,25,37,39,40);(7,11,12,10)(8,13,9,14)(16,23)(17,27)(22,38)(24,34)(26,30)(28,42)(29,43)(41,45)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283617);