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×#13322 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1488 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)(9,21,26,52)(10,27,30,49)(11,31,36,47)(12,19,13,18)(15,38,42,23)(16,37,44,29)(17,34,48,35)(22,53,56,70)(24,45,25,51)(28,46,33,43)(32,40,58,41)(39,62,69,65)(50,59,67,55)(54,60,57,68)(61,66,64,63);(3,4)(5,6)(7,9,14,26)(8,15,20,42)(10,22,30,56)(11,32,36,58)(12,25,13,24)(16,39,44,69)(17,45,48,51)(18,41,19,40)(21,47,52,31)(23,34,38,35)(27,60,49,68)(28,62,33,65)(29,59,37,55)(43,70,46,53)(50,61,67,64)(54,66,57,63);(4,6)(7,10)(8,16)(9,22)(11,33)(12,37)(13,29)(14,30)(15,39)(17,46)(18,49)(19,27)(20,44)(21,50)(23,57)(24,59)(25,55)(26,56)(28,36)(31,64)(32,65)(34,63)(35,66)(38,54)(40,60)(41,68)(42,69)(43,48)(45,53)(47,61)(51,70)(52,67)(58,62)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406025);