Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24)(26,30)(27,33)(32,37)(35,39);(1,2)(3,7)(4,6)(5,8)(9,10,13,11)(12,14,16,15)(17,19)(18,23)(20,21)(22,24)(28,36)(32,37)(35,39)(38,40);(3,8)(5,7)(10,11)(12,16)(18,20)(19,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,21,24,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,3,4,8)(2,5,6,7)(9,12,13,16)(10,14,11,15)(17,21)(18,19)(20,24)(22,23)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fgi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5394663);