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×#28702 |
| 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)(11,12,15,19)(13,22,24,17)(14,26,28,37)(16,21,31,20)(18,30,36,29)(23,39,40,35)(25,33,42,34)(27,32,41,38);(3,5,7,6)(4,8,10,9)(11,13,16,25,15,24,31,42)(12,17,21,34,19,22,20,33)(14,27,30,40,28,41,29,23)(18,35,26,38,36,39,37,32);(11,14,16,29,15,28,31,30)(12,18,20,26,19,36,21,37)(13,23,25,41,24,40,42,27)(17,32,33,35,22,38,34,39);(3,6,7,5)(4,9,10,8)(12,19)(17,22)(18,36)(20,21)(26,37)(32,38)(33,34)(35,39)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373492);