Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,22)(14,28)(15,20)(16,33)(17,21)(18,24)(19,31)(23,38)(25,35)(26,42)(27,40)(29,37)(30,36)(32,39)(34,41);(3,5,6,10)(4,7,8,9)(11,12)(13,23)(14,19)(15,33)(16,20)(17,21)(18,25)(22,38)(24,35)(26,34)(27,37)(28,31)(29,40)(30,39)(32,36)(41,42);(11,13,17,27)(12,18,21,34)(14,29,32,38)(15,26,16,25)(19,35,36,42)(20,37,33,23)(22,39,40,28)(24,30,41,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406166);