Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19054 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(5,9)(8,10)(11,12)(13,23)(14,28)(15,20)(16,21)(17,22)(18,26)(19,30)(24,32)(25,33)(27,36)(29,34)(31,35)(37,39)(38,41)(40,42);(5,9)(7,10)(11,13)(12,18)(14,24)(15,25)(16,26)(17,27)(19,32)(20,23)(21,36)(22,33)(28,41)(29,38)(30,39)(31,40)(34,42)(35,37);(3,4)(5,10)(6,8)(7,9)(11,14,17,31)(12,19,22,34)(13,24,27,40)(15,30,16,29)(18,32,33,42)(20,35,21,28)(23,37,36,41)(25,39,26,38);(11,15,17,16)(12,20,22,21)(13,25,27,26)(14,29,31,30)(18,23,33,36)(19,28,34,35)(24,38,40,39)(32,41,42,37)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258653);