Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#23542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,6)(11,12,17,22)(13,23,27,34)(14,28,30,38)(15,21,16,20)(18,26,35,25)(19,32,37,31)(24,40,41,33)(29,36,42,39);(7,8)(9,10)(11,13,17,27)(12,18,22,35)(14,29,30,42)(15,26,16,25)(19,36,37,39)(20,34,21,23)(24,31,41,32)(28,40,38,33);(4,6)(8,10)(11,14,15,31,17,30,16,32)(12,19,20,28,22,37,21,38)(13,24,25,42,27,41,26,29)(18,33,23,39,35,40,34,36);(14,30)(19,37)(24,41)(28,38)(29,42)(31,32)(33,40)(36,39)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339155);