Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13402 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13403 |
| 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 | |
1038 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)(5,15,20,35)(6,21,26,56)(7,27,29,12)(8,30,32,54)(10,19,36,17)(11,37,41,24)(13,42,44,28)(16,46,49,66)(18,34,53,64)(22,57,47,65)(23,55,25,43)(31,40,58,39)(33,61,63,51)(38,60,62,48)(45,50,59,52);(3,5)(4,10)(6,22)(7,19)(8,18)(9,20)(11,38)(12,35)(13,34)(14,36)(15,27)(16,25)(17,29)(21,46)(23,49)(24,51)(26,47)(28,52)(30,59)(31,60)(32,53)(33,40)(37,61)(39,63)(41,62)(42,50)(43,65)(44,64)(45,54)(48,58)(55,57)(56,66);(3,6,9,26)(4,11,14,41)(5,16)(7,25,29,23)(8,31,32,58)(10,33)(12,40,27,39)(13,43,44,55)(15,38)(17,22)(18,51)(19,47)(20,49)(21,54,56,30)(24,42,37,28)(34,46)(35,62)(36,63)(45,65)(48,52)(50,60)(53,61)(57,59)(64,66)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406235);