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×#57694 |
| 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 | |
4354 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);(3,4,7,5)(6,9,10,8)(11,12)(13,19)(14,21)(15,23)(16,17)(18,24)(20,22)(25,26)(27,28,33,30)(29,31,34,37)(32,35,41,38)(36,39,40,42);(4,5)(8,9)(12,17)(14,19)(15,24)(20,26)(27,29,33,34)(28,31,30,37)(32,40,41,36)(35,42,38,39);(11,13,16,21)(12,14,17,19)(15,20,24,26)(18,22,23,25)(27,30,33,28)(29,31,34,37)(32,38,41,35)(36,39,40,42);(3,5,7,4)(6,9,10,8)(11,14)(12,13)(15,22)(16,19)(17,21)(18,20)(23,26)(24,25)(27,31)(28,34)(29,30)(32,39)(33,37)(35,40)(36,38)(41,42);(3,6,7,10)(4,8,5,9)(11,15,16,24)(12,18,17,23)(13,20,21,26)(14,22,19,25)(27,32)(28,35)(29,36)(30,38)(31,39)(33,41)(34,40)(37,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009068);