Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#29724 |
| 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 | |
2858 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);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,22)(19,24)(20,23)(25,34)(26,36)(27,35)(28,37)(29,39)(30,38)(31,41)(32,42)(33,40)(43,45)(44,46);(8,11)(12,14)(15,17,20,27)(16,21,23,35)(18,28,30,44)(19,31,33,26)(22,37,38,46)(24,41,40,36)(25,32,43,29)(34,42,45,39);(4,6)(7,9)(8,12)(10,13)(11,14)(15,18,20,30)(16,22,23,38)(17,25,27,43)(19,32,33,29)(21,34,35,45)(24,42,40,39)(26,44,31,28)(36,46,41,37);(9,13)(12,14)(16,23)(21,35)(22,38)(24,40)(34,45)(36,41)(37,46)(39,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340172);