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×#52761 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5168 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)(11,12,14,16)(13,17,18,15)(19,20,24,25)(21,26,28,23)(22,30,31,27)(29,32,34,33);(3,4,8,9)(5,6,7,10)(13,18)(15,17)(20,25)(21,28)(22,31)(32,33);(3,5,8,7)(4,10,9,6)(11,13,14,18)(12,15,16,17)(19,21)(20,26)(22,29)(23,25)(24,28)(27,33)(30,32)(31,34);(3,6,8,10)(4,5,9,7)(11,12)(13,15)(14,16)(17,18)(19,22,24,31)(20,27,25,30)(21,29,28,34)(23,32,26,33);(3,7,8,5)(4,10,9,6)(19,23,24,26)(20,21,25,28)(22,32,31,33)(27,29,30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999833);