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×#36151 |
| 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 | |
7970 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);(8,12)(13,14)(15,16,19,18)(17,21,22,20);(4,6)(7,8)(9,13)(10,14)(11,12)(15,17,19,22)(16,20,18,21);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,16,19,18)(17,20,22,21);(4,6)(7,9,11,10)(8,13,12,14);(3,4)(5,6)(7,10,11,9)(8,14,12,13)(15,18,19,16)(17,21,22,20)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536977);