Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5456 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);(3,4,8,10)(5,9,6,7)(19,20)(21,27)(22,26)(23,24)(25,28)(29,34)(30,33)(31,32);(3,5)(4,9)(6,8)(7,10)(11,12,15,16)(13,18,17,14)(20,23)(21,28)(26,31)(29,30);(4,10)(5,6)(12,16)(13,17)(19,21)(20,25)(22,30)(23,27)(24,28)(26,34)(29,32)(31,33);(3,6,8,5)(4,7,10,9)(11,13,15,17)(12,14,16,18)(19,22,24,32)(20,26,23,31)(21,29,28,30)(25,33,27,34);(3,7)(4,5)(6,10)(8,9)(11,14)(12,17)(13,16)(15,18)(19,23,24,20)(21,27,28,25)(22,31,32,26)(29,34,30,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614357);