Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42328 |
| 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 | |
4496 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);(3,4,7,5)(6,9,10,8)(27,28)(29,34)(30,38)(31,36)(32,33)(35,40)(37,39)(41,42);(3,5,7,4)(6,9,10,8)(12,16)(14,22)(17,19)(20,26)(28,33)(29,36)(35,38)(37,42);(3,6)(4,8)(5,9)(7,10)(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24)(27,29,32,36)(28,34,33,31)(30,37,40,42)(35,41,38,39);(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,30,32,40)(28,35,33,38)(29,37,36,42)(31,39,34,41);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,34)(28,36,33,29)(30,39,40,41)(35,42,38,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547984);