Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46221 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4616 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);(4,8)(9,10)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24)(27,28,32,35)(29,36,33,30)(31,34,39,41)(37,42,40,38);(12,14)(17,19)(18,22)(24,25)(29,33)(30,36)(31,39)(34,41);(3,4,7,8)(5,9,6,10)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,29,32,33)(28,30,35,36)(31,37,39,40)(34,38,41,42);(3,5,7,6)(4,9,8,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,30)(28,33)(29,35)(31,38)(32,36)(34,40)(37,41)(39,42);(3,6,7,5)(4,10,8,9)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,39)(28,34,35,41)(29,37,33,40)(30,38,36,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609960);