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×#50109 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4768 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);(11,12,15,14)(13,17,18,16)(19,20,24,27)(21,23,29,25)(22,26,32,34)(28,31,30,33);(4,7)(5,9)(19,21,24,29)(20,25,27,23)(22,28,32,30)(26,33,34,31);(3,4,6,7)(5,8,9,10)(11,13,15,18)(12,16,14,17)(19,21)(20,25)(22,30)(23,27)(24,29)(26,31)(28,32)(33,34);(3,4)(5,8)(6,7)(9,10)(11,12,15,14)(13,16,18,17)(19,22)(20,26)(21,28)(23,31)(24,32)(25,33)(27,34)(29,30);(3,5)(4,8)(6,9)(7,10)(11,14,15,12)(13,17,18,16)(19,23,24,25)(20,21,27,29)(22,31,32,33)(26,28,34,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612422);