Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26990 |
| 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 | |
2238 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)(24,38)(26,34)(29,33)(32,37)(35,39)(36,42)(40,41);(3,5,6,10)(4,7,8,9)(11,13,16,25,17,14,28,26)(12,18,21,19,22,34,23,27)(15,24,31,39,32,29,42,40)(20,33,30,35,36,38,37,41);(4,8)(7,9)(11,14)(12,19)(13,17)(15,29)(16,26)(18,23)(20,35)(21,34)(22,27)(24,32)(25,28)(30,38)(31,40)(33,37)(36,41)(39,42);(4,8)(7,9)(11,15,28,42,17,32,16,31)(12,20,23,37,22,36,21,30)(13,24,26,40,14,29,25,39)(18,33,27,41,34,38,19,35)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350915);