Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35461 |
| 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 | |
5490 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);(4,9)(5,10)(12,16)(17,18)(19,20,24,23)(21,25,27,29)(22,26,32,31)(28,33,30,34);(3,4,8,9)(5,6,10,7)(11,12)(13,17)(14,18)(15,16)(20,23)(21,27)(26,31)(28,30);(3,5)(4,6)(7,9)(8,10)(19,21)(20,25)(22,30)(23,29)(24,27)(26,34)(28,32)(31,33);(3,6,8,7)(4,10,9,5)(11,13,15,14)(12,17,16,18)(19,22,24,32)(20,26,23,31)(21,28,27,30)(25,33,29,34);(3,7,8,6)(4,5,9,10)(11,14,15,13)(12,18,16,17)(19,23,24,20)(21,29,27,25)(22,31,32,26)(28,34,30,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536025);