Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32865 |
| 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×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1604 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);(3,4,6,9)(5,8,10,7)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,26)(23,25)(27,31)(28,33)(29,32)(30,34);(3,5)(4,7)(6,10)(8,9)(12,15)(16,18)(19,21,23,29)(20,24,25,32)(22,27,30,28)(26,31,34,33);(3,5)(4,8)(6,10)(7,9)(12,15)(13,17)(21,27)(22,30)(24,31)(26,34)(28,29)(32,33);(5,10)(7,8)(11,13,14,17)(12,16,15,18)(20,25)(24,32)(26,34)(31,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538755);