Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33628 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,21)(14,25)(15,30)(16,18)(17,22)(19,28)(20,27)(23,41)(24,38)(26,36)(29,37)(31,35)(32,40)(33,42)(34,39);(3,5,6,10)(4,7,8,9);(11,13,16,24)(12,17,18,32)(14,26,29,42)(15,31,27,23)(19,35,34,41)(20,36,30,33)(21,37,38,25)(22,39,40,28);(12,18)(17,32)(19,34)(20,30)(21,38)(25,37)(33,36)(35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530199);