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×#62327 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,20)(14,27)(15,23)(16,22)(17,21)(18,26)(19,30)(24,38)(25,33)(28,36)(29,35)(31,37)(32,42)(34,39)(40,41);(3,5)(4,7)(6,10)(8,9)(11,13,16,26)(12,18,22,20)(14,24,29,40)(15,31,17,25)(19,32,36,34)(21,37,23,33)(27,41,35,38)(28,42,30,39);(11,14,17,30,16,29,15,28)(12,19,23,27,22,36,21,35)(13,24,25,39,26,40,31,42)(18,32,33,41,20,34,37,38);(5,10)(7,9)(11,13)(12,20)(14,24)(15,31)(16,26)(17,25)(18,22)(19,34)(21,33)(23,37)(27,38)(28,42)(29,40)(30,39)(32,36)(35,41);(4,8)(7,9)(11,15,16,17)(12,21,22,23)(13,25,26,31)(14,28,29,30)(18,33,20,37)(19,35,36,27)(24,39,40,42)(32,41,34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484816);