Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#11029 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,6)(11,12,16,22,17,20,15,21)(13,23,26,34,27,18,25,35)(14,28,31,37,32,19,30,38)(24,33,40,42,41,39,29,36);(4,6)(7,8)(9,10)(11,13,17,27)(12,18,20,23)(14,29,32,40)(15,26,16,25)(19,36,28,42)(21,35,22,34)(24,31,41,30)(33,38,39,37);(8,10)(11,14,17,32)(12,19,20,28)(13,24,27,41)(15,31,16,30)(18,33,23,39)(21,38,22,37)(25,40,26,29)(34,42,35,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264735);