Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1930 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);(3,4,6,8)(5,9,10,7)(11,12,16,22)(13,23,26,19)(14,27,29,41)(15,31,17,21)(18,32,34,25)(20,30,36,28)(24,38,40,33)(35,39,37,42);(11,13,15,25,16,26,17,32)(12,18,21,23,22,34,31,19)(14,24,28,39,29,40,30,42)(20,33,27,37,36,38,41,35);(3,5,6,10)(4,7,8,9)(11,13,16,26)(12,19,22,23)(14,24,29,40)(15,25,17,32)(18,31,34,21)(20,35,36,37)(27,33,41,38)(28,39,30,42);(11,14,17,30,16,29,15,28)(12,20,31,41,22,36,21,27)(13,24,32,42,26,40,25,39)(18,33,19,35,34,38,23,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350692);