Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28297 |
| 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 | |
1936 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)(13,18)(14,26)(15,31)(16,20)(17,21)(19,28)(22,35)(23,40)(24,33)(25,34)(27,41)(29,36)(30,37)(32,38)(39,42);(3,5,7,6)(4,8,10,9)(11,13,16,24,17,25,15,23)(12,18,20,33,21,34,31,40)(14,27,29,39,30,38,28,22)(19,35,26,41,36,42,37,32);(11,14,17,30)(12,19,21,36)(13,22,25,39)(15,29,16,28)(18,32,34,41)(20,37,31,26)(23,27,24,38)(33,42,40,35);(3,6,7,5)(4,9,10,8)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373273);