Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#32734 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,9)(5,8,10,7)(11,12)(13,15)(14,16)(17,18)(19,20,23,25)(21,24,29,32)(22,26,30,34)(27,31,28,33);(3,5)(4,7)(6,10)(8,9)(11,13)(12,15)(14,17)(16,18)(19,21)(20,24)(22,27)(23,29)(25,32)(26,31)(28,30)(33,34);(3,5)(4,8)(6,10)(7,9)(12,16)(15,18)(21,27)(22,30)(24,31)(26,34)(28,29)(32,33);(5,10)(7,8)(13,17)(15,18)(20,25)(24,32)(26,34)(31,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538441);