Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53035 |
| 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 | |
4886 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)(5,7)(11,12)(13,17)(14,15)(16,18)(19,20,24,25)(21,26,27,22)(23,31,32,28)(29,30,34,33);(3,4,8,9)(5,6,7,10)(12,15)(13,18)(20,25)(22,26)(28,31)(30,33);(3,5,8,7)(4,10,9,6)(11,12)(13,16)(14,15)(17,18)(19,21)(20,26)(22,25)(23,29)(24,27)(28,33)(30,31)(32,34);(3,6,8,10)(4,5,9,7)(11,13,14,18)(12,16,15,17)(19,22,24,26)(20,27,25,21)(23,30,32,33)(28,34,31,29);(3,7,8,5)(4,10,9,6)(19,23,24,32)(20,28,25,31)(21,29,27,34)(22,30,26,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000101);