Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1254 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,28,32)(27,31,35,30)(29,33,36,39)(34,38,40,37);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(2,6)(7,8)(25,27,29,34,28,35,36,40)(26,30,33,37,32,31,39,38);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,35)(30,31)(34,40)(37,38);(1,4,5,3)(2,8,6,7)(11,19)(14,17)(18,24)(22,23)(25,27,36,40,28,35,29,34)(26,31,39,37,32,30,33,38)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,d^2=fgi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6748835);