Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#29789 |
| 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 | |
2038 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);(4,7)(8,10)(11,12,15,20,17,22,16,21)(13,18,24,33,25,34,31,38)(14,26,28,37,30,36,29,19)(23,35,27,41,39,42,40,32);(3,4,6,7)(5,8,9,10)(11,13)(12,18)(14,27)(15,24)(16,31)(17,25)(19,35)(20,33)(21,38)(22,34)(23,29)(26,41)(28,39)(30,40)(32,36)(37,42);(3,5)(4,8)(6,9)(7,10)(11,14,17,30)(12,19,22,37)(13,23,25,39)(15,29,16,28)(18,32,34,41)(20,36,21,26)(24,40,31,27)(33,42,38,35);(5,9)(8,10)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340423);