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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(4,8)(11,15)(13,16)(17,18,20,22)(19,23,24,21)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40);(1,2,5,6)(3,8,7,4)(9,10,12,14)(11,13,15,16)(19,24)(21,23)(26,28)(31,33)(32,36)(38,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2)(3,4)(5,6)(7,8)(17,19,20,24)(18,21,22,23)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=f,d^2=i,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, 4801795);