Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62329 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6446 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)(11,12,17,21)(13,18,24,34)(14,25,28,39)(15,20,16,30)(19,26,35,27)(22,36,38,41)(23,32,29,33)(31,40,42,37);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,22)(15,29)(16,23)(17,24)(19,31)(20,33)(21,34)(25,36)(26,40)(27,37)(28,38)(30,32)(35,42)(39,41);(11,14,15,26,17,28,16,27)(12,19,30,39,21,35,20,25)(13,22,29,40,24,38,23,37)(18,31,32,41,34,42,33,36);(5,10)(7,9)(11,13)(12,18)(14,22)(15,29)(16,23)(17,24)(19,31)(20,33)(21,34)(25,36)(26,40)(27,37)(28,38)(30,32)(35,42)(39,41);(4,8)(7,9)(11,15,17,16)(12,20,21,30)(13,23,24,29)(14,26,28,27)(18,32,34,33)(19,25,35,39)(22,37,38,40)(31,41,42,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484815);