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×#58641 |
| 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 | |
4386 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);(4,8)(6,9)(27,28,33,36)(29,34,30,31)(32,40,41,35)(37,39,38,42);(3,4)(5,9)(6,10)(7,8)(12,16)(14,22)(17,19)(20,26)(27,29,33,30)(28,31,36,34)(32,38,41,37)(35,42,40,39);(4,8)(5,10)(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(3,5)(4,6)(7,10)(8,9)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,31,33,34)(28,30,36,29)(32,39,41,42)(35,38,40,37);(3,6)(4,5)(7,9)(8,10)(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,32,33,41)(28,35,36,40)(29,37,30,38)(31,39,34,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012072);