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×#57513 |
| 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 | |
4506 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);(12,16)(13,19)(18,23)(20,25)(27,28)(29,35)(30,38)(31,32)(33,37)(34,39)(36,41)(40,42);(3,4,7,9)(5,6,10,8)(12,16)(17,22)(18,23)(24,26)(28,31)(30,39)(33,35)(36,40);(3,5,7,10)(4,8,9,6)(11,12,15,16)(13,17,19,22)(14,18,21,23)(20,24,25,26)(27,29)(28,33)(30,40)(31,35)(32,37)(34,42)(36,39)(38,41);(3,5)(4,6)(7,10)(8,9)(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,30)(28,34)(29,36)(31,38)(32,39)(33,41)(35,42)(37,40);(3,6)(4,5)(7,8)(9,10)(11,14,15,21)(12,18,16,23)(13,20,19,25)(17,24,22,26)(27,31,32,28)(29,35,37,33)(30,38,39,34)(36,42,40,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008213);