Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#31274 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2666 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)(13,15)(14,16)(17,18)(19,20)(21,26)(22,27)(23,25)(24,28)(29,34)(30,32)(31,33);(3,5)(4,7)(6,10)(8,9)(11,13,14,17)(12,15,16,18)(20,25)(21,29)(22,24)(26,33)(30,31)(32,34);(12,16)(15,18)(19,21)(20,26)(22,29)(23,30)(24,31)(25,32)(27,34)(28,33);(4,8)(5,10)(12,16)(15,18)(19,22,23,24)(20,27,25,28)(21,29,30,31)(26,34,32,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411454);