Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#459 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1616 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)(6,14)(9,15)(10,17)(11,18)(12,16)(19,20)(21,25)(22,29)(23,31)(24,27)(26,28)(30,33)(32,34);(3,4,7,9)(5,11,13,17)(6,15,14,8)(10,16,18,12)(19,21,24,23)(20,25,27,31)(22,28,30,32)(26,33,34,29);(4,9)(8,15)(10,18)(11,17)(19,22)(20,26)(21,28)(23,32)(24,30)(25,33)(27,34)(29,31);(3,5,7,13)(4,10,9,18)(6,16,14,12)(8,17,15,11)(19,21,24,23)(20,25,27,31)(22,28,30,32)(26,33,34,29)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265519);