Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| 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 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,14,19,15,20,27,17)(13,22,24,31,25,18,26,34)(16,21,28,36,29,37,42,32)(23,39,40,33,41,35,30,38);(3,5,6,9)(4,7,8,10)(12,17)(13,23,25,41)(14,27)(16,28)(18,33,22,38)(19,20)(24,30,26,40)(29,42)(31,35,34,39)(32,36);(4,8)(7,10)(11,13,14,24,15,25,27,26)(12,18,19,34,20,22,17,31)(16,30,28,23,29,40,42,41)(21,33,36,35,37,38,32,39)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406763);