Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13523 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13524 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2092 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
712 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);(3,4,7,13,32,15,36,48,10,17,8,14,9,16,33,49)(5,18,20,52,21,50,34,40,23,54,59,39,22,46,30,11)(6,24,26,44,37,47,35,12,29,42,27,43,28,51,31,45)(19,38,55,60,25,61,62,66,57,64,63,65,56,41,58,53);(3,5,9,21,10,23,32,22)(4,11,16,52,17,40,15,39)(6,25,28,57,29,56,37,19)(7,30,33,20,8,34,36,59)(12,41,44,38,45,61,43,64)(13,46,49,18,14,50,48,54)(24,60,51,66,42,65,47,53)(26,55,31,62,27,63,35,58);(3,6,10,29)(4,12,17,45)(5,19,23,57)(7,31,8,35)(9,37,32,28)(11,38,40,64)(13,47,14,51)(15,44,16,43)(18,53,54,66)(20,58,59,62)(21,56,22,25)(24,49,42,48)(26,36,27,33)(30,55,34,63)(39,61,52,41)(46,60,50,65)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406713);