Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13450 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,6)(11,12,19,26)(13,27,32,42)(14,33,30,40)(15,23,34,24)(16,22,17,37)(18,38,28,25)(20,36,41,29)(21,35,39,31);(7,8)(9,10)(11,13,19,32)(12,20,26,41)(14,34,30,15)(16,29,17,36)(18,35,28,31)(21,22,39,37)(23,27,24,42)(25,33,38,40);(4,6)(8,10)(11,14,15,13,28,29,17,35,19,30,34,32,18,36,16,31)(12,21,22,41,38,42,24,40,26,39,37,20,25,27,23,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406363);