Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7366 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);(3,4,6,8)(5,9,10,7)(11,12,15,17)(13,21,23,34)(14,19,25,31)(16,28,27,20)(18,29,30,24)(22,35,36,42)(26,39,38,33)(32,40,41,37);(3,5,6,10)(4,7,8,9)(12,17)(18,30)(19,31)(20,28)(21,34)(32,41)(33,39)(35,42);(11,13,15,23)(12,18,17,30)(14,22,25,36)(16,29,27,24)(19,32,31,41)(20,34,28,21)(26,40,38,37)(33,42,39,35);(11,14)(12,19)(13,22)(15,25)(16,26)(17,31)(18,32)(20,33)(21,35)(23,36)(24,37)(27,38)(28,39)(29,40)(30,41)(34,42);(14,25)(19,31)(22,36)(26,38)(32,41)(33,39)(35,42)(37,40)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482197);