Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,14)(11,15,16,12)(19,23)(22,24)(25,26,30,31)(27,35,36,32)(28,33,34,29)(37,38,40,39);(1,2,4,6)(3,5,8,7)(11,16)(12,15)(18,21)(22,24)(26,31)(27,36)(28,34)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,36)(26,32,31,35)(28,37,34,40)(29,38,33,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(25,28)(26,33)(27,37)(29,31)(30,34)(32,39)(35,38)(36,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,33)(26,34,31,28)(27,38,36,39)(32,40,35,37)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4776408);