Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33768 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 74 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 74 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,17,22)(13,18,24,39)(14,27,30,61)(15,31,35,60)(16,36,33,21)(19,42,44,28)(20,45,47,34)(23,49,50,72)(25,55,53,71)(26,58,54,41)(29,62,37,46)(32,48,63,43)(38,64,65,51)(40,68,67,57)(52,73,59,69)(56,70,74,66);(3,5)(4,7)(6,10)(8,9)(11,13,17,24)(12,18,22,39)(14,23,30,50)(15,25,35,53)(16,26,33,54)(19,38,44,65)(20,40,47,67)(21,41,36,58)(27,49,61,72)(28,51,42,64)(29,52,37,59)(31,55,60,71)(32,56,63,74)(34,57,45,68)(43,66,48,70)(46,69,62,73);(5,10)(7,9)(11,14,17,30)(12,19,22,44)(13,23,24,50)(15,32,35,63)(16,37,33,29)(18,38,39,65)(20,46,47,62)(21,48,36,43)(25,56,53,74)(26,59,54,52)(27,60,61,31)(28,45,42,34)(40,69,67,73)(41,70,58,66)(49,71,72,55)(51,68,64,57);(4,8)(7,9)(13,24)(18,39)(23,50)(25,53)(26,54)(38,65)(40,67)(41,58)(49,72)(51,64)(52,59)(55,71)(56,74)(57,68)(66,70)(69,73)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530497);