Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13441 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 70 points:
(1,2);(3,4)(5,6)(7,8,13,18)(9,19,23,51)(10,24,29,42)(11,30,34,49)(12,35,32,17)(14,37,40,21)(15,26,44,27)(16,45,47,33)(20,52,55,68)(22,56,36,46)(25,50,53,41)(28,64,61,58)(31,48,66,39)(38,59,67,54)(43,63,57,62)(60,69,65,70);(7,9,13,23)(8,14,18,40)(10,25,29,53)(11,31,34,66)(12,36,32,22)(15,41,44,50)(16,46,47,56)(17,48,35,39)(19,49,51,30)(20,26,55,27)(21,45,37,33)(24,38,42,67)(28,65,61,60)(43,70,57,69)(52,58,68,64)(54,63,59,62);(4,6)(7,10,11,26,32,61,45,63,13,29,34,27,12,28,33,62)(8,15,16,42,35,57,49,58,18,44,47,24,17,43,30,64)(9,20,21,53,36,59,66,60,23,55,37,25,22,54,31,65)(14,38,19,50,48,68,56,69,40,67,51,41,39,52,46,70)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406353);