Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28136 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,18)(14,27)(15,20)(16,21)(17,22)(19,30)(23,36)(24,33)(25,34)(26,35)(28,41)(29,37)(31,38)(32,39)(40,42);(3,5,6,10)(4,7,8,9)(11,13,16,25,17,26,15,24)(12,18,21,34,22,35,20,33)(14,28,29,40,31,39,30,23)(19,36,27,41,37,42,38,32);(11,14)(12,19)(13,23)(15,29)(16,30)(17,31)(18,32)(20,27)(21,38)(22,37)(24,28)(25,39)(26,40)(33,36)(34,42)(35,41);(4,8)(7,9)(11,15,17,16)(12,20,22,21)(13,24,26,25)(14,29,31,30)(18,33,35,34)(19,27,37,38)(23,28,40,39)(32,36,41,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366648);