Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13455 |
| 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 | |
810 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);(3,4,7,9)(5,10,6,8)(11,12)(13,17)(14,18)(15,20)(16,22)(19,27)(21,28)(23,36)(24,33)(25,31)(26,32)(29,35)(30,37)(34,40)(38,42)(39,41);(3,5,7,6)(4,8,9,10)(11,13)(12,17)(14,24)(15,23)(16,26)(18,33)(19,30)(20,36)(21,31)(22,32)(25,28)(27,37)(29,38)(34,41)(35,42)(39,40);(3,6,7,5)(4,8,9,10)(12,18)(13,23)(14,27)(15,28)(17,30)(19,21)(20,34)(22,35)(24,38)(25,26)(29,40)(31,42)(32,41)(33,36)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406396);