Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#21181 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#121 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2642 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);(3,4,6,5)(7,8)(9,19)(10,24)(11,16)(12,17)(13,18)(14,22)(15,26)(20,34)(21,29)(23,32)(25,31)(27,30)(28,37)(33,36)(35,38);(3,5,6,4)(7,9)(8,14)(10,20)(11,21)(12,22)(13,23)(15,28)(16,19)(17,32)(18,29)(24,33)(25,35)(26,36)(27,37)(30,38)(31,34);(7,10,12,26,13,27,11,25)(8,15,16,24,18,31,17,30)(9,20,22,36,23,37,21,35)(14,28,19,33,29,34,32,38);(7,11,13,12)(8,16,18,17)(9,21,23,22)(10,25,27,26)(14,19,29,32)(15,24,31,30)(20,35,37,36)(28,33,34,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279104);