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×#57895 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4474 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)(5,8)(6,7)(9,10)(11,12,16,15)(13,19,20,17)(14,22,23,18)(21,24,26,25)(27,28)(29,34)(30,31)(32,35)(33,36)(37,41)(38,39)(40,42);(4,6)(5,9)(13,20)(17,19)(21,26)(24,25)(27,29,33,31)(28,34,36,30)(32,39,40,37)(35,38,42,41);(3,5,7,9)(4,8,6,10)(11,13,16,20)(12,17,15,19)(14,21,23,26)(18,24,22,25)(27,30)(28,29)(31,36)(32,38)(33,34)(35,37)(39,42)(40,41);(3,5)(4,8)(6,10)(7,9)(11,14,16,23)(12,18,15,22)(13,21,20,26)(17,24,19,25)(27,31,33,29)(28,34,36,30)(32,39,40,37)(35,41,42,38);(3,6,7,4)(5,10,9,8)(11,15,16,12)(13,19,20,17)(14,22,23,18)(21,25,26,24)(27,32,33,40)(28,35,36,42)(29,37,31,39)(30,38,34,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008675);