Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26995 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2228 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)(13,19)(14,27)(15,30)(16,23)(17,22)(18,25)(20,31)(21,28)(24,39)(26,34)(29,33)(32,38)(35,40)(36,41)(37,42);(3,5,6,10)(4,7,8,9)(11,13,16,25,17,14,28,26)(12,18,21,19,22,34,23,27)(15,24,31,40,32,29,42,41)(20,33,30,35,37,39,38,36);(4,8)(7,9)(11,14)(12,19)(13,17)(15,24)(16,26)(18,23)(20,36)(21,34)(22,27)(25,28)(29,32)(30,33)(31,40)(35,37)(38,39)(41,42);(4,8)(7,9)(11,15,16,31,17,32,28,42)(12,20,21,30,22,37,23,38)(13,24,25,40,14,29,26,41)(18,33,19,35,34,39,27,36)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350922);