Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54854 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5244 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);(3,4,6,7)(5,8,9,10)(19,20,24,25)(21,26,29,32)(22,27,31,34)(23,30,33,28);(5,9)(8,10)(11,12,15,16)(13,14,17,18)(20,25)(21,29)(22,31)(23,33);(4,7)(8,10)(12,16)(13,17)(19,21)(20,26)(22,28)(23,27)(24,29)(25,32)(30,31)(33,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,17)(16,18)(19,22)(20,27)(21,30)(23,32)(24,31)(25,34)(26,33)(28,29);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,18)(16,17)(19,23,24,33)(20,28,25,30)(21,27,29,34)(22,32,31,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009756);