Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13544 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #2101 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4)(5,17,20,45)(6,12,24,39)(7,13)(8,16)(9,14)(10,15)(11,19,35,31)(18,47)(21,41,52,65)(22,54)(23,37,32,46)(25,38,57,64)(26,40)(27,44)(28,59,58,34)(29,42,48,66)(30,43)(33,51)(36,53)(49,61)(50,60)(55,63)(56,62);(3,5,10,31,8,20,7,19)(4,11,13,17,16,35,15,45)(6,22,32,53,24,56,23,55)(9,28,30,21,27,58,26,52)(12,36,37,54,39,63,46,62)(14,41,40,59,44,65,43,34)(18,48,51,57,50,29,49,25)(33,42,47,64,61,66,60,38);(3,6,7,23,8,24,10,32)(4,12,13,37,16,39,15,46)(5,18,19,49,20,50,31,51)(9,29,26,57,27,48,30,25)(11,33,17,47,35,61,45,60)(14,42,40,64,44,66,43,38)(21,53,28,22,52,55,58,56)(34,62,41,36,59,54,65,63)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406755);