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×#42774 |
| 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 | |
4350 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);(5,9)(8,10)(11,12,16,19)(13,17,20,15)(14,18,24,26)(21,25,22,23)(27,28)(29,34)(30,38)(31,36)(32,33)(35,40)(37,42)(39,41);(3,4,7,6)(5,8,9,10)(13,20)(15,17)(21,22)(23,25)(28,33)(31,41)(34,42)(35,38);(3,5,7,9)(4,8,6,10)(11,13)(12,17)(14,22)(15,19)(16,20)(18,23)(21,24)(25,26)(27,29)(28,34)(30,39)(31,35)(32,37)(33,42)(36,40)(38,41);(3,6,7,4)(5,10,9,8)(11,14)(12,18)(13,21)(15,23)(16,24)(17,25)(19,26)(20,22)(27,30,32,40)(28,35,33,38)(29,36,37,39)(31,34,41,42);(11,15,16,17)(12,13,19,20)(14,23,24,25)(18,21,26,22)(27,31,32,41)(28,36,33,39)(29,35,37,38)(30,34,40,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548457);