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×#55081 |
| 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 | |
4500 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);(11,12)(13,17)(14,21)(15,16)(18,22)(19,25)(20,23)(24,26)(27,28,33,36)(29,34,30,31)(32,40,41,35)(37,39,38,42);(3,4,7,8)(5,6,9,10)(12,16)(17,23)(18,21)(24,25)(27,29,33,30)(28,31,36,34)(32,38,41,37)(35,42,40,39);(4,8)(5,9)(11,13,15,20)(12,17,16,23)(14,19,22,26)(18,24,21,25)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(3,5)(4,6)(7,9)(8,10)(11,12)(13,17)(14,18)(15,16)(19,24)(20,23)(21,22)(25,26)(27,31,33,34)(28,30,36,29)(32,39,41,42)(35,38,40,37);(3,6)(4,5)(7,10)(8,9)(11,14,15,22)(12,18,16,21)(13,19,20,26)(17,24,23,25)(27,32,33,41)(28,35,36,40)(29,37,30,38)(31,39,34,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009614);