Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#49587 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8376 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);(3,4,5,6)(7,8)(9,14)(10,13)(11,12);(7,9,12,10)(8,13,11,14)(16,19)(17,21);(3,4)(5,6)(9,10)(13,14)(15,16,18,19)(17,20,21,22);(7,10,12,9)(8,13,11,14)(15,16)(17,20)(18,19)(21,22);(7,11,12,8)(9,13,10,14)(15,17)(16,20)(18,21)(19,22)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612816);