Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,10)(7,8)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24)(27,28)(29,34)(30,36)(31,33)(32,35)(37,41)(38,42)(39,40);(4,8)(5,10)(12,14)(17,19)(18,22)(24,25)(27,29,33,30)(28,34,31,36)(32,38,40,37)(35,42,39,41);(3,5)(4,6)(7,10)(8,9)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,30,33,29)(28,34,31,36)(32,38,40,37)(35,41,39,42);(3,5)(4,6)(7,10)(8,9)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,31,33,28)(29,36,30,34)(32,39,40,35)(37,42,38,41);(3,6)(4,5)(7,9)(8,10)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,32,33,40)(28,35,31,39)(29,37,30,38)(34,41,36,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007089);