Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#29674 |
| 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 | |
3332 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)(15,16,20,24)(17,21,26,35)(18,28,30,38)(19,31,33,23)(22,32,37,29)(25,39,40,46)(27,43,42,36)(34,44,45,41);(3,4)(5,6)(8,11)(12,14)(15,17,20,26)(16,21,24,35)(18,25,30,40)(19,27,33,42)(22,34,37,45)(23,36,31,43)(28,39,38,46)(29,41,32,44);(4,6)(9,13)(12,14)(15,18,20,30)(16,22,24,37)(17,25,26,40)(19,32,33,29)(21,34,35,45)(23,38,31,28)(27,44,42,41)(36,46,43,39);(7,9,10,13)(8,12,11,14)(17,26)(21,35)(25,40)(27,42)(34,45)(36,43)(39,46)(41,44)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342202);