Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62350 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3547 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6462 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);(3,4)(5,9)(6,8)(7,10)(11,12,16,20)(13,17,22,30)(14,24,26,39)(15,28,25,19)(18,32,33,27)(21,34,37,42)(23,38,35,31)(29,40,41,36);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,21)(15,23)(16,22)(18,29)(19,31)(20,30)(24,34)(25,35)(26,37)(27,36)(28,38)(32,40)(33,41)(39,42);(11,14,25,32,16,26,15,27)(12,18,28,39,20,33,19,24)(13,21,35,40,22,37,23,36)(17,29,38,42,30,41,31,34);(5,10)(7,9)(11,13)(12,17)(14,21)(15,23)(16,22)(18,29)(19,31)(20,30)(24,34)(25,35)(26,37)(27,36)(28,38)(32,40)(33,41)(39,42);(4,8)(7,9)(13,22)(14,26)(17,30)(18,33)(23,35)(24,39)(27,32)(31,38)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484838);