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×#36180 |
| 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 | |
8730 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)(9,12)(10,13)(11,14)(15,16,19,21)(17,18,20,22);(7,9,10,14)(8,11,13,12)(15,17)(16,18)(19,20)(21,22);(3,4,5,6)(7,8,10,13)(9,12,14,11)(15,18)(16,20)(17,21)(19,22);(7,8)(9,11)(10,13)(12,14)(15,17)(16,18)(19,20)(21,22);(7,9)(8,12)(10,14)(11,13)(15,18)(16,20)(17,21)(19,22)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537013);