Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#39068 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,7)(8,10)(19,20)(21,25)(22,29)(23,24)(26,30)(27,33)(28,31)(32,34);(4,7)(5,9)(12,15)(13,17)(20,24)(25,31)(26,29)(32,33);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,21,23,28)(20,25,24,31)(22,27,30,34)(26,32,29,33);(19,22,23,30)(20,26,24,29)(21,27,28,34)(25,32,31,33);(3,5)(4,8)(6,9)(7,10)(11,13,14,17)(12,16,15,18)(19,20)(21,25)(22,26)(23,24)(27,32)(28,31)(29,30)(33,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543651);