Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#28072 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1952 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)(5,9)(6,8)(7,10)(11,12,14,16)(13,15,17,18)(19,20,24,28)(21,29,31,26)(22,25,32,34)(23,33,30,27);(3,5,6,10)(4,7,8,9)(11,13)(12,15)(14,17)(16,18)(19,21,30,32,24,31,23,22)(20,25,33,26,28,34,27,29);(4,8)(7,9)(13,17)(15,18)(21,31)(22,32)(25,34)(26,29);(4,8)(7,9)(19,22,23,31,24,32,30,21)(20,26,27,25,28,29,33,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364586);