Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#28766 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2162 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);(3,4,7,10)(5,9,6,8)(11,12,15,19)(13,22,24,17)(14,26,29,37)(16,21,28,20)(18,31,36,30)(23,39,41,35)(25,33,40,34)(27,32,42,38);(3,5,7,6)(4,8,10,9)(11,13,16,25,15,24,28,40)(12,17,21,34,19,22,20,33)(14,27,31,41,29,42,30,23)(18,35,26,38,36,39,37,32);(3,5,7,6)(4,9,10,8)(11,14,28,31,15,29,16,30)(12,18,21,37,19,36,20,26)(13,23,40,27,24,41,25,42)(17,32,34,39,22,38,33,35);(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, 6374069);