Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24860 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,7,10)(5,9,6,8)(11,12,17,22,15,20,16,21)(13,23,32,34,25,18,26,38)(14,27,31,19,29,37,30,36)(24,40,42,39,41,35,28,33);(3,5,7,6)(4,8,10,9)(11,13,15,25)(12,18,20,23)(14,28,29,42)(16,26,17,32)(19,35,36,40)(21,34,22,38)(24,30,41,31)(27,39,37,33);(11,14,15,29)(12,19,20,36)(13,24,25,41)(16,31,17,30)(18,33,23,39)(21,37,22,27)(26,42,32,28)(34,40,38,35);(3,6,7,5)(4,9,10,8)(12,20)(18,23)(19,36)(21,22)(27,37)(33,39)(34,38)(35,40)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352858);