Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20178 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3330 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,8)(6,15)(9,16)(10,11)(12,17)(13,18)(19,20,23,27)(21,24,28,32)(22,25,30,26)(29,33,34,31);(3,4)(5,12)(6,10)(7,11)(8,15)(9,14)(13,16)(17,18)(19,21)(20,24)(22,29)(23,28)(25,33)(26,31)(27,32)(30,34);(3,5,7,14)(4,9,11,12)(6,13,15,18)(8,16,10,17)(20,25)(21,28)(22,30)(24,31)(26,27)(32,33);(4,10)(6,15)(8,11)(9,17)(12,16)(13,18)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266725);