Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 46 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 46 points:
(1,2);(7,8)(9,13)(10,11)(12,14)(15,16,20,24)(17,21,26,34)(18,28,31,43)(19,32,29,23)(22,36,37,30)(25,38,40,46)(27,42,39,35)(33,44,45,41);(3,4)(5,6)(8,11)(12,13)(15,17)(16,21)(18,25)(19,27)(20,26)(22,33)(23,35)(24,34)(28,38)(29,39)(30,41)(31,40)(32,42)(36,44)(37,45)(43,46);(4,6)(15,18,29,36,20,31,19,30)(16,22,32,43,24,37,23,28)(17,25,39,44,26,40,27,41)(21,33,42,46,34,45,35,38);(7,9,10,14)(8,12,11,13)(17,26)(21,34)(25,40)(27,39)(33,45)(35,42)(38,46)(41,44)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356324);