Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#30257 |
| 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 | |
2690 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)(9,12)(10,11)(13,14)(15,16,20,25,21,26,19,24)(17,22,28,36,29,37,33,39)(18,30,31,38,32,40,34,23)(27,41,43,46,44,45,42,35);(3,4,5,6)(8,11)(12,14)(15,17)(16,22)(18,27)(19,33)(20,28)(21,29)(23,35)(24,39)(25,36)(26,37)(30,41)(31,43)(32,44)(34,42)(38,46)(40,45);(4,6)(7,9)(8,12)(10,13)(11,14)(15,18,21,32)(16,23,26,38)(17,27,29,44)(19,31,20,34)(22,35,37,46)(24,30,25,40)(28,42,33,43)(36,45,39,41);(9,13)(12,14)(15,19,21,20)(16,24,26,25)(17,28,29,33)(18,31,32,34)(22,36,37,39)(23,30,38,40)(27,42,44,43)(35,45,46,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340978);