Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13435 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1077 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)(5,15,19,39)(6,20,24,51)(7,25,28,12)(8,29,31,13)(10,26,34,17)(11,35,38,23)(16,44,21,52)(18,48,30,43)(22,55,57,50)(27,40,61,59)(32,62,36,65)(33,47,41,60)(37,56,58,64)(42,54,66,46)(45,63,53,49);(3,5,9,19)(4,10,14,34)(6,21,24,16)(7,26,28,17)(8,30,31,18)(11,36,38,32)(12,39,25,15)(13,41,29,33)(20,49,51,63)(22,45,57,53)(23,54,35,46)(27,47,61,60)(37,42,58,66)(40,43,59,48)(44,55,52,50)(56,65,64,62);(3,6,7,22,9,24,28,57)(4,11,12,37,14,38,25,58)(5,16,17,45,19,21,26,53)(8,23,27,56,31,35,61,64)(10,32,15,42,34,36,39,66)(13,20,40,50,29,51,59,55)(18,46,47,65,30,54,60,62)(33,63,43,44,41,49,48,52)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406345);