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×#56976 |
| 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 | |
5038 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)(5,10)(6,9)(7,8)(11,12,15,17)(13,14,18,16)(19,20,24,22)(21,25,29,27)(23,30,31,26)(28,33,34,32);(3,5)(4,9)(6,8)(7,10)(11,13,15,18)(12,16,17,14)(20,22)(25,27)(26,30)(32,33);(5,6)(9,10)(12,17)(14,16)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33);(3,6,8,5)(4,9,7,10)(11,14,15,16)(12,13,17,18)(19,22,24,20)(21,27,29,25)(23,30,31,26)(28,33,34,32);(3,7,8,4)(5,9,6,10)(11,13)(12,16)(14,17)(15,18)(19,23,24,31)(20,26,22,30)(21,28,29,34)(25,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006856);