Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62990 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10)(11,15)(12,22,24,16)(13,18)(14,17)(19,26,32,34)(20,28)(21,27)(23,35,36,29)(25,30)(31,37,39,40)(33,38);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,26)(17,27)(18,28)(22,34)(23,31)(24,32)(25,33)(29,37)(30,38)(35,40)(36,39);(2,5)(3,7)(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,22)(18,29)(20,31)(21,32)(25,36)(27,34)(28,37)(30,35)(33,39)(38,40);(1,3)(2,6)(4,7)(5,8)(10,17)(15,27)(16,22)(18,30)(26,34)(28,38)(29,35)(37,40);(2,5)(3,7)(9,13,14,25)(10,18,17,30)(11,20,21,33)(12,23,24,36)(15,28,27,38)(16,29,22,35)(19,31,32,39)(26,37,34,40)Polycyclic presentation:
2^9|a^2=f,b^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503390);