Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#36181 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(4,6)(7,8,11,13)(9,10,12,14)(16,20)(17,22);(7,9)(8,10)(11,12)(13,14)(15,16)(17,21)(18,22)(19,20);(3,4,5,6)(7,10)(8,12)(9,13)(11,14)(15,17,19,22)(16,21,20,18);(7,9)(8,10)(11,12)(13,14)(15,18,19,21)(16,22,20,17);(7,10)(8,12)(9,13)(11,14)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537014);