Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1555 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)(3,5)(4,6)(7,8)(10,14)(11,12)(19,20)(23,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(2,3)(6,7)(9,10)(11,15)(12,16)(13,14)(18,22)(23,24)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(1,2,5,3)(4,7,8,6)(9,11,13,12)(10,15,14,16)(17,18,21,22)(19,23,20,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,15,14,16)(17,19,21,20)(18,23,22,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,4,5,8)(2,6,3,7)(9,10)(11,16)(12,15)(13,14)(17,20,21,19)(18,24,22,23)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)Polycyclic presentation:
2^9|b^2=fg,c^2=i,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4613158);