Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25691 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2952 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);(3,4,6,7)(5,9,10,8)(11,12,17,21)(13,18,24,35)(14,25,27,38)(15,29,16,20)(19,31,37,28)(22,36,39,42)(23,34,30,33)(26,41,40,32);(4,7)(8,9)(11,13)(12,18)(14,26)(15,23)(16,30)(17,24)(19,36)(20,33)(21,35)(22,28)(25,41)(27,40)(29,34)(31,39)(32,38)(37,42);(11,14,15,28,17,27,16,31)(12,19,20,25,21,37,29,38)(13,22,23,40,24,39,30,26)(18,32,33,36,35,41,34,42);(3,5,6,10)(4,8,7,9)(14,27)(19,37)(22,39)(25,38)(26,40)(28,31)(32,41)(36,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357295);