Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#24525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2668 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)(11,12,15,20,17,22,16,21)(13,18,24,33,26,35,25,34)(14,27,31,19,30,38,29,37)(23,39,28,32,41,36,40,42);(3,5,6,10)(4,7,8,9)(11,13)(12,18)(14,28)(15,24)(16,25)(17,26)(19,36)(20,33)(21,34)(22,35)(23,29)(27,32)(30,40)(31,41)(37,39)(38,42);(11,14)(12,19)(13,23)(15,29)(16,31)(17,30)(18,32)(20,27)(21,38)(22,37)(24,40)(25,28)(26,41)(33,39)(34,36)(35,42);(4,8)(7,9)(11,15,17,16)(12,20,22,21)(13,24,26,25)(14,29,30,31)(18,33,35,34)(19,27,37,38)(23,40,41,28)(32,39,42,36)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359220);