Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3245 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,12)(6,17)(7,10)(8,11)(9,14)(13,16)(15,18)(19,20)(21,27)(22,30)(23,26)(24,29)(25,32)(28,34)(31,33);(3,5,8,16)(4,9,11,18)(6,15,7,14)(10,13,17,12)(24,33)(27,34);(3,4)(5,13)(6,10)(7,17)(8,11)(9,14)(12,16)(15,18)(19,21)(20,24)(22,31)(23,28)(25,34)(26,33)(27,30)(29,32)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139868);