Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53255 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4538 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);(5,9)(8,10)(11,12,15,18)(13,16,20,23)(14,17,21,24)(19,22,25,26)(27,28)(29,31)(30,34)(32,40)(33,36)(35,41)(37,42)(38,39);(4,7)(8,10)(14,21)(17,24)(19,25)(22,26)(27,29,33,30)(28,34,36,31)(32,37,41,38)(35,42,40,39);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,26)(27,30,33,29)(28,34,36,31)(32,38,41,37)(35,42,40,39);(3,5)(4,8)(6,9)(7,10)(11,13,15,20)(12,16,18,23)(14,19,21,25)(17,22,24,26)(27,31,33,34)(28,29,36,30)(32,39,41,42)(35,37,40,38);(3,4)(5,8)(6,7)(9,10)(11,14)(12,17)(13,19)(15,21)(16,22)(18,24)(20,25)(23,26)(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, 8012999);