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×#56448 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5054 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);(12,15)(13,17)(19,20)(21,27)(22,29)(23,31)(24,25)(26,32)(28,30)(33,34);(3,4)(5,9)(6,8)(7,10)(12,15)(16,18)(20,25)(22,27)(23,32)(28,34);(3,4,6,8)(5,7,10,9)(11,12)(13,16)(14,15)(17,18)(19,21,24,29)(20,22,25,27)(23,28,32,34)(26,30,31,33);(3,4)(5,7)(6,8)(9,10)(11,13,14,17)(12,16,15,18)(19,22)(20,21)(23,30)(24,27)(25,29)(26,28)(31,34)(32,33);(3,5)(4,7)(6,10)(8,9)(19,23,24,32)(20,26,25,31)(21,28,29,34)(22,30,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006174);