Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1590 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)(5,8)(6,7)(9,10)(11,12,14,19)(13,22,23,17)(15,18,26,33)(16,29,27,21)(20,35,34,28)(24,36,39,30)(25,32,40,38)(31,41,37,42);(4,7)(8,10)(11,13)(12,17)(14,23)(15,24)(16,25)(18,30)(19,22)(20,31)(21,32)(26,39)(27,40)(28,41)(29,38)(33,36)(34,37)(35,42);(12,18)(15,28)(16,27)(17,30)(19,33)(20,21)(22,36)(24,41)(25,40)(26,35)(29,34)(31,32)(37,38)(39,42);(3,5,6,9)(4,8,7,10)(13,23)(17,22)(24,39)(25,40)(30,36)(31,37)(32,38)(41,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530286);