Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37413 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,6)(5,9,10,8)(12,16)(14,21)(17,22)(20,25)(27,28,32,35)(29,36,30,33)(31,39,40,34)(37,41,38,42);(12,16)(13,19)(18,23)(20,25)(27,29,32,30)(28,33,35,36)(31,37,40,38)(34,41,39,42);(3,5,7,10)(4,8,6,9)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,30,32,29)(28,33,35,36)(31,38,40,37)(34,41,39,42);(3,6,7,4)(5,9,10,8)(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,28)(29,36)(30,33)(31,34)(32,35)(37,42)(38,41)(39,40);(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,31,32,40)(28,34,35,39)(29,37,30,38)(33,41,36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541995);