Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10422 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,5)(7,8,11,14)(9,12,10,13)(15,16)(17,18)(19,21)(20,22);(3,4)(5,6)(7,9,11,10)(8,12,14,13)(15,17,20,21)(16,18,22,19);(3,5,6,4)(8,13)(9,10)(12,14)(16,17)(18,19)(21,22)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146791);