Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60666 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6442 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)(14,19,25,30)(16,28,27,20)(18,31,29,24)(22,35,36,42)(26,39,38,33)(32,41,40,37);(3,5,6,10)(4,7,8,9)(12,17)(18,29)(19,30)(20,28)(21,34)(32,40)(33,39)(35,42);(11,13,16,24,15,23,27,31)(12,18,20,21,17,29,28,34)(14,22,26,37,25,36,38,41)(19,32,33,35,30,40,39,42);(11,14)(12,19)(13,22)(15,25)(16,26)(17,30)(18,32)(20,33)(21,35)(23,36)(24,37)(27,38)(28,39)(29,40)(31,41)(34,42);(14,25)(19,30)(22,36)(26,38)(32,40)(33,39)(35,42)(37,41)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482472);