Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 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);(3,4,7,10)(5,9,6,8)(11,12,15,20)(13,22,24,18)(14,27,29,37)(16,31,17,21)(19,32,36,30)(23,35,40,42)(25,34,26,39)(28,38,41,33);(3,5,7,6)(4,8,10,9)(11,13,16,25,15,24,17,26)(12,18,31,39,20,22,21,34)(14,28,32,23,29,41,30,40)(19,35,27,33,36,42,37,38);(11,14,16,30,15,29,17,32)(12,19,21,27,20,36,31,37)(13,23,25,28,24,40,26,41)(18,33,34,42,22,38,39,35);(3,6,7,5)(4,9,10,8)(12,20)(18,22)(19,36)(21,31)(27,37)(33,38)(34,39)(35,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373491);