Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(5,9)(8,10)(12,16)(17,18)(19,20)(21,27)(22,30)(23,24)(25,29)(26,31)(28,32)(33,34);(4,7)(5,9)(20,24)(25,27)(26,30)(32,33);(3,4)(5,8)(6,7)(9,10)(11,12,15,16)(13,17,14,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,33)(31,34);(3,5,6,9)(4,8,7,10)(11,13,15,14)(12,17,16,18)(19,20)(21,25)(22,26)(23,24)(27,29)(28,32)(30,31)(33,34);(3,4)(5,8)(6,7)(9,10)(11,14,15,13)(12,18,16,17)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609780);