Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | Q8×#391 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(17,18,20,22)(19,24,21,23);(17,19,20,21)(18,23,22,24);(2,4)(5,6,8,7)(11,15)(13,16)(17,21,20,19)(18,23,22,24);(1,2,3,4)(5,7,8,6)(9,10,12,14)(11,13,15,16);(9,11,12,15)(10,13,14,16)(17,22,20,18)(19,24,21,23)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426291);