Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C22×#2370 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2);(3,4);(5,6,8,7)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,24)(21,22);(5,7,8,6)(10,12)(14,15)(18,22)(20,23);(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,23,22,20);(9,12,13,10)(11,15,16,14)(17,20,21,23)(18,24,22,19)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532571);