Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13473 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2067 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
736 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);(3,4,11,16,10,17,9,36)(5,18,38,53,23,41,22,59)(6,24,37,49,29,45,28,48)(7,30,35,15,8,32,19,14)(12,39,42,33,43,54,52,21)(13,31,46,26,47,34,50,27)(20,55,60,61,57,40,56,66)(25,62,64,65,63,51,58,44);(3,5,19,54,9,22,8,33,10,23,35,39,11,38,7,21)(4,12,30,53,16,42,15,41,17,43,32,59,36,52,14,18)(6,25,31,20,28,58,27,56,29,63,34,57,37,64,26,60)(13,44,48,55,46,62,24,61,47,65,49,40,50,51,45,66);(3,6,10,29)(4,13,17,47)(5,20,23,57)(7,31,8,34)(9,37,11,28)(12,40,43,55)(14,48,15,49)(16,50,36,46)(18,51,41,62)(19,26,35,27)(21,58,33,64)(22,60,38,56)(24,30,45,32)(25,39,63,54)(42,61,52,66)(44,53,65,59)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406514);