Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
| direct product | C2×#34255 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4,6,5)(8,10)(12,13)(15,16,20,18)(17,22,19,21);(7,8,11,10)(9,13,14,12)(16,18)(21,22);(3,5,6,4)(15,17,20,19)(16,21,18,22);(7,9,11,14)(8,12,10,13)(15,18,20,16)(17,22,19,21);(7,10,11,8)(9,13,14,12)(15,19,20,17)(16,22,18,21)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532658);