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×#42289 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,7,10)(5,8,6,9)(11,12,16,14)(13,17,19,21)(15,18,23,22)(20,24,25,26)(27,28,32,33)(29,31,36,34)(30,38,40,35)(37,41,42,39);(3,5,7,6)(4,8,10,9)(13,19)(15,23)(17,21)(18,22)(28,33)(29,36)(35,38)(37,42);(3,4)(5,8)(6,9)(7,10)(11,13)(12,17)(14,21)(15,20)(16,19)(18,24)(22,26)(23,25)(27,29,32,36)(28,34,33,31)(30,37,40,42)(35,41,38,39);(3,5,7,6)(4,8,10,9)(11,14,16,12)(13,17,19,21)(15,22,23,18)(20,24,25,26)(27,30,32,40)(28,35,33,38)(29,37,36,42)(31,39,34,41);(3,6,7,5)(4,9,10,8)(11,15)(12,18)(13,20)(14,22)(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, 7547831);