Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#2064 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2);(3,4);(9,10,13,20,15,19,14,18)(11,17,21,16,12,23,22,24);(5,6)(7,8)(9,11,13,21,15,12,14,22)(10,16,18,17,19,24,20,23);(6,8)(9,12,15,11)(10,17,19,23)(13,21,14,22)(16,20,24,18)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339310);