Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
2364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,13)(14,16)(17,18,21,24)(19,20,23,22)(28,35)(32,37)(33,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(26,30)(31,36)(32,37)(38,40);(17,19)(18,22)(20,24)(21,23)(25,26,29,30)(27,31,34,36)(28,32,35,37)(33,38,39,40);(1,3)(2,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,19)(18,20)(21,23)(22,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,4)(2,7)(3,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,20)(18,23)(19,24)(21,22)(25,28,29,35)(26,32,30,37)(27,33,34,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=hi,e^2=h,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, 4803049);