Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| 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 | |
1034 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)(6,21,25,56)(7,27,29,12)(8,30,32,61)(10,19,36,17)(11,37,40,24)(13,42,44,28)(16,47,22,58)(18,34,53,65)(23,57,26,43)(31,41,60,39)(33,51,38,63)(45,59,66,49)(46,50,62,52)(48,64,54,55);(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)(21,55)(23,48)(24,59)(26,54)(28,52)(30,62)(31,63)(32,53)(33,40)(37,49)(39,45)(41,66)(42,50)(43,47)(44,65)(46,61)(51,60)(56,64)(57,58);(3,6,7,23,9,25,29,26)(4,11,12,39,14,40,27,41)(5,16,17,48,20,22,19,54)(8,31,28,37,32,60,42,24)(10,33,15,45,36,38,35,66)(13,43,30,56,44,57,61,21)(18,51,50,49,53,63,52,59)(34,58,46,64,65,47,62,55)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406220);