Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26651 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2249 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)(11,12,15,21,16,22,17,23)(13,24,27,35,28,36,29,18)(14,30,31,38,32,19,33,20)(25,39,26,34,40,37,41,42);(11,13,16,28)(12,18,22,35)(14,26,32,41)(15,29,17,27)(19,37,30,39)(20,34,38,42)(21,36,23,24)(25,33,40,31);(4,7)(8,10)(11,14)(12,19)(13,25)(15,31)(16,32)(17,33)(18,34)(20,21)(22,30)(23,38)(24,37)(26,27)(28,40)(29,41)(35,42)(36,39);(3,5,6,9)(4,8,7,10)(11,14)(12,20)(13,26)(15,33)(16,32)(17,31)(18,34)(19,21)(22,38)(23,30)(24,39)(25,27)(28,41)(29,40)(35,42)(36,37)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346798);