Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1810 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);(3,4,7,9)(5,10,6,8)(11,12,15,16)(13,18,14,17)(19,20,23,26)(21,24,29,33)(22,25,30,34)(27,31,28,32);(3,5,7,6)(4,8,9,10)(12,16)(17,18)(19,21,23,29)(20,24,26,33)(22,27,30,28)(25,31,34,32);(3,6,7,5)(4,8,9,10)(11,13,15,14)(12,17,16,18)(21,27)(22,30)(24,31)(25,34)(28,29)(32,33);(4,9)(8,10)(11,14,15,13)(12,18,16,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357478);