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×#62439 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6802 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);(4,7)(5,9)(11,12)(13,22)(14,26)(15,20)(16,21)(17,31)(18,25)(19,29)(23,37)(24,34)(27,35)(28,41)(30,33)(32,42)(36,38)(39,40);(3,4)(5,8)(6,7)(9,10)(11,13)(12,18)(14,23)(15,30)(16,25)(17,24)(19,32)(20,34)(21,22)(26,40)(27,42)(28,39)(29,38)(31,33)(35,36)(37,41);(11,14,15,27,16,28,17,29)(12,19,31,41,21,35,20,26)(13,23,30,42,25,39,24,38)(18,32,33,37,22,36,34,40);(4,7)(8,10)(11,15,16,17)(12,20,21,31)(13,24,25,30)(14,27,28,29)(18,33,22,34)(19,26,35,41)(23,38,39,42)(32,37,36,40);(3,5,6,9)(4,8,7,10)(12,21)(13,25)(19,35)(20,31)(23,39)(24,30)(26,41)(38,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486401);