Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19991 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1828 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);(3,4)(5,12)(6,8)(7,11)(9,14)(10,15)(13,17)(16,18)(19,20)(21,24)(22,25)(23,27)(26,30)(28,31)(29,33)(32,34);(4,8)(5,13)(6,15)(9,12)(10,11)(14,16)(19,21,23,29)(20,24,27,33)(22,28,30,34)(25,31,26,32);(20,25)(22,30)(24,31)(26,27)(28,34)(32,33);(3,5,7,14)(4,9,11,12)(6,16,15,13)(8,17,10,18)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265578);