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×#18977 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3547 | 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)(5,14)(6,11)(7,12)(8,17)(9,13)(10,15)(16,18)(19,20,25,28)(21,31,27,34)(22,32,26,33)(23,30,24,29);(3,5,9,6)(4,10,13,18)(7,15,8,16)(11,17,14,12)(19,21,25,27)(20,26,28,22)(23,33,24,32)(29,34,30,31);(3,6,9,5)(4,11,13,14)(7,15,8,16)(10,17,18,12)(19,22,25,26)(20,27,28,21)(23,31,24,34)(29,32,30,33);(3,7,9,8)(4,12,13,17)(5,15,6,16)(10,11,18,14)(20,28)(22,26)(29,30)(31,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256149);