Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33124 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12,16,20)(13,21,25,35)(14,26,28,18)(15,29,22,19)(17,30,32,24)(23,37,39,34)(27,33,36,41)(31,38,40,42);(3,5,6,9)(4,7,8,10)(11,13,22,30,16,25,15,24)(12,17,29,35,20,32,19,21)(14,23,36,42,28,39,27,38)(18,31,41,37,26,40,33,34);(4,8)(7,10);(11,14,15,27,16,28,22,36)(12,18,19,33,20,26,29,41)(13,23,24,38,25,39,30,42)(17,31,21,34,32,40,35,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536899);