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×#57699 |
| 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 | |
4718 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);(3,4,7,9)(5,6,10,8)(11,12)(13,17)(14,22)(15,16)(18,23)(19,20)(21,26)(24,25)(27,28,32,33)(29,30,37,34)(31,39,40,35)(36,41,42,38);(3,5,7,10)(4,8,9,6)(12,16)(13,19)(14,23)(24,26)(28,33)(30,34)(31,40)(36,42);(11,13,15,19)(12,17,16,20)(14,21,23,25)(18,24,22,26)(27,29,32,37)(28,34,33,30)(31,36,40,42)(35,41,39,38);(3,5)(4,6)(7,10)(8,9)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,30,32,34)(28,29,33,37)(31,38,40,41)(35,36,39,42);(3,6)(4,5)(7,8)(9,10)(11,14)(12,18)(13,21)(15,23)(16,22)(17,24)(19,25)(20,26)(27,31)(28,35)(29,36)(30,38)(32,40)(33,39)(34,41)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008721);