Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2077 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);(3,4,9,14)(5,15,19,37)(6,20,23,40)(7,24,28,12)(8,29,31,13)(10,25,34,17)(11,30,36,22)(16,43,46,65)(18,48,50,42)(21,53,26,52)(27,39,59,56)(32,61,62,66)(33,47,60,57)(35,55,38,58)(41,49,64,45)(44,51,54,63);(3,5)(4,10)(6,16)(7,25)(8,18)(9,19)(11,32)(12,37)(13,33)(14,34)(15,24)(17,28)(20,51)(21,54)(22,45)(23,46)(26,44)(27,57)(29,60)(30,49)(31,50)(35,64)(36,62)(38,41)(39,48)(40,63)(42,56)(43,53)(47,59)(52,65)(55,61)(58,66);(3,6,9,23)(4,11,14,36)(5,16,19,46)(7,26,28,21)(8,30,31,22)(10,32,34,62)(12,38,24,35)(13,40,29,20)(15,41,37,64)(17,44,25,54)(18,49,50,45)(27,55,59,58)(33,63,60,51)(39,52,56,53)(42,65,48,43)(47,61,57,66)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406135);