Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10665 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #399 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,5,8,7,10,6,9)(11,12,15,20,16,21,17,31)(13,22,24,34,26,38,25,18)(14,27,29,37,30,19,32,36)(23,39,28,42,40,35,41,33);(3,4)(5,9)(6,8)(7,10)(11,13,17,24,16,26,15,25)(12,18,31,22,21,34,20,38)(14,28,32,40,30,41,29,23)(19,35,37,33,27,39,36,42);(4,8)(5,6)(9,10)(11,14)(12,19)(13,23)(15,29)(16,30)(17,32)(18,33)(20,36)(21,27)(22,39)(24,28)(25,41)(26,40)(31,37)(34,42)(35,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151080);