Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46219 |
| 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 | |
4964 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);(4,6)(7,9)(12,16)(17,18)(19,20,24,27)(21,28,25,22)(23,26,31,33)(29,34,32,30);(3,4,8,6)(5,7,10,9)(21,25)(22,28)(23,31)(26,33);(3,5,8,10)(4,9,6,7)(11,12,15,16)(13,17,14,18)(19,21,24,25)(20,22,27,28)(23,29,31,32)(26,30,33,34);(3,6,8,4)(5,7,10,9)(11,13,15,14)(12,17,16,18)(19,22)(20,25)(21,27)(23,30)(24,28)(26,32)(29,33)(31,34);(3,7)(4,5)(6,10)(8,9)(11,14,15,13)(12,18,16,17)(19,23,24,31)(20,26,27,33)(21,29,25,32)(22,30,28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609939);