Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13392 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #2033 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1044 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)(5,15)(6,21)(7,27)(8,30)(9,14)(10,17)(11,24)(12,29)(13,28)(16,46)(18,35)(19,36)(20,37)(22,56)(23,54)(25,55)(26,42)(31,41)(32,39)(33,60)(34,62)(38,59)(40,58)(43,61)(44,48)(45,51)(47,63)(49,65)(50,64)(52,53)(57,66);(3,5)(4,10)(6,22)(7,19)(8,31)(9,20)(11,34)(12,37)(13,41)(14,36)(15,27)(16,25)(17,29)(18,33)(21,53)(23,47)(24,48)(26,52)(28,49)(30,45)(32,50)(35,43)(38,66)(39,44)(40,64)(42,46)(51,61)(54,56)(55,63)(57,58)(59,62)(60,65);(3,6,7,23,9,25,29,26)(4,11,12,38,14,40,27,39)(5,16,17,47,20,22,19,52)(8,32,28,58,33,59,61,24)(10,34,15,44,36,64,37,66)(13,42,30,55,43,54,60,21)(18,50,49,48,31,62,51,57)(35,56,45,63,41,46,65,53)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406224);