Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18930 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18976 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3554 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2810 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)(6,9)(7,11)(8,16)(10,14)(13,17)(15,18)(19,20,24,27)(21,29,30,33)(22,25,23,28)(26,31,34,32);(3,5)(4,10)(6,13)(7,14)(8,17)(9,15)(11,18)(12,16)(20,25)(22,23)(26,33)(27,28)(29,34)(31,32);(3,6,9,4)(5,13,15,10)(7,11,8,16)(12,14,18,17)(19,21,24,30)(20,26,27,34)(22,32,23,31)(25,33,28,29);(3,7,9,8)(4,11,6,16)(5,14,15,17)(10,18,13,12)(20,27)(21,30)(25,28)(31,32)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256148);