Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61052 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1911 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)(3,5)(4,6)(7,8)(9,10,15,20,16,12,14,19)(11,21,24,22,26,36,25,17)(13,18,30,35,31,27,29,34)(23,33,28,37,39,40,38,32);(1,2)(3,7)(4,6)(5,8)(9,11,16,26)(10,17,12,22)(13,28,31,38)(14,24,15,25)(18,33,27,40)(19,21,20,36)(23,29,39,30)(32,34,37,35);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,22)(13,27)(14,20)(15,19)(17,26)(18,31)(21,25)(23,37)(24,36)(28,40)(29,35)(30,34)(32,39)(33,38);(3,8)(5,7)(9,13,15,30,16,31,14,29)(10,18,20,35,12,27,19,34)(11,23,24,28,26,39,25,38)(17,32,21,33,22,37,36,40);(2,6)(5,7)(9,14,16,15)(10,19,12,20)(11,24,26,25)(13,29,31,30)(17,21,22,36)(18,34,27,35)(23,28,39,38)(32,33,37,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6963118);