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×#25529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3272 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)(5,9)(6,8)(7,10)(11,12)(13,18)(14,25)(15,29)(16,20)(17,21)(19,28)(22,36)(23,34)(24,35)(26,41)(27,38)(30,33)(31,37)(32,40)(39,42);(3,5,6,10)(4,7,8,9)(11,13,17,24)(12,18,21,35)(14,26,27,40)(15,23,16,30)(19,36,37,42)(20,33,29,34)(22,31,39,28)(25,41,38,32);(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);(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, 6359420);