Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C22×#1723 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(6,9)(7,11)(10,12)(13,14,17,20)(15,21,24,28)(16,25,22,19)(18,26,27,23);(5,6,7,9,8,10,11,12);(6,10)(9,12)(13,15,22,26,17,24,16,23)(14,18,25,28,20,27,19,21)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264291);