Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1042 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)(5,15,20,36)(6,21,25,58)(7,26,29,12)(8,30,33,57)(10,19,37,17)(11,38,41,24)(13,43,46,28)(16,49,22,59)(18,35,31,44)(23,61,27,45)(32,42,62,40)(34,63,39,54)(47,60,64,51)(48,53,65,55)(50,66,52,56);(3,5)(4,10)(6,22)(7,19)(8,31)(9,20)(11,39)(12,36)(13,44)(14,37)(15,26)(16,25)(17,29)(18,33)(21,56)(23,50)(24,51)(27,52)(28,53)(30,48)(32,54)(34,41)(35,46)(38,60)(40,64)(42,47)(43,55)(45,49)(57,65)(58,66)(59,61)(62,63);(3,6,9,25)(4,11,14,41)(5,16)(7,27,29,23)(8,32,33,62)(10,34)(12,42,26,40)(13,45,46,61)(15,47)(17,52)(18,54)(19,50)(20,22)(21,57,58,30)(24,43,38,28)(31,63)(35,49)(36,64)(37,39)(44,59)(48,66)(51,55)(53,60)(56,65)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406238);