Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13339 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1524 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);(7,8,14,20)(9,21,26,56)(10,27,33,46)(11,34,39,52)(12,19,13,18)(15,40,44,23)(16,29,49,30)(17,37,53,38)(22,41,60,69)(24,50,25,55)(28,65,59,45)(31,51,32,64)(35,42,62,43)(36,47,67,48)(54,57,70,58)(61,63,66,68);(3,4)(5,6)(7,9,14,26)(8,15,20,44)(10,28,33,59)(11,35,39,62)(12,25,13,24)(16,45,49,65)(17,50,53,55)(18,43,19,42)(21,52,56,34)(22,32,60,31)(23,37,40,38)(27,63,46,68)(29,57,30,58)(36,66,67,61)(41,48,69,47)(51,70,64,54);(4,6)(7,10)(8,16)(9,22)(11,36)(12,32)(13,31)(14,33)(15,41)(17,51)(18,48)(19,47)(20,49)(21,54)(23,61)(24,59)(25,28)(26,60)(27,34)(29,37)(30,38)(35,58)(39,67)(40,66)(42,65)(43,45)(44,69)(46,52)(50,63)(53,64)(55,68)(56,70)(57,62)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406072);