Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | Q8×#554 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(9,10,13,12)(11,15,16,14)(17,18,21,20)(19,23,24,22);(10,12)(14,15)(17,19,21,24)(18,22,20,23);(9,11)(10,14)(12,15)(13,16);(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,22,24,23)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606841);