Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,20)(11,21,26,32)(12,18,29,34)(13,19,14,30)(16,25,33,24)(17,27,35,28)(22,38,40,31)(23,37,39,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,22)(13,24)(14,25)(15,26)(17,31)(18,36)(19,32)(20,33)(21,30)(23,27)(28,39)(29,40)(34,37)(35,38);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,31)(18,30)(19,34)(20,35)(21,36)(23,25)(24,39)(26,40)(32,37)(33,38);(3,8)(5,7)(9,12)(10,18)(11,23)(13,27)(14,28)(15,29)(16,31)(17,30)(19,35)(20,34)(21,37)(22,25)(24,40)(26,39)(32,36)(33,38);(2,6)(5,7)(9,13,15,14)(10,19,20,30)(11,24,26,25)(12,27,29,28)(16,32,33,21)(17,18,35,34)(22,23,40,39)(31,36,38,37)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962128);