Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C4×#2398 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3846 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);(9,10,13,15)(11,16,12,14)(17,18,20,22)(19,23,24,21);(5,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,20,24)(18,21,22,23);(6,8)(18,22)(21,23);(9,12,13,11)(10,14,15,16)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421237);