Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#21220 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,6)(7,8,11,16,12,15,17,29)(9,13,20,25,19,26,23,30)(10,14,22,28,21,27,31,37)(18,24,33,36,32,35,34,38);(7,9,17,20,12,19,11,23)(8,13,16,30,15,26,29,25)(10,18,31,33,21,32,22,34)(14,24,28,38,27,35,37,36);(3,4,5,6)(7,10)(8,14)(9,18)(11,22)(12,21)(13,24)(15,27)(16,28)(17,31)(19,32)(20,33)(23,34)(25,36)(26,35)(29,37)(30,38);(10,21)(14,27)(18,32)(22,31)(24,35)(28,37)(33,34)(36,38)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279174);