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 | |
2109 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)(11,22)(12,18)(13,19)(14,20)(15,21)(16,26)(17,29)(23,31)(24,38)(25,33)(27,37)(28,34)(30,36)(32,40)(35,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,23)(13,25)(14,26)(15,27)(17,31)(18,35)(19,22)(20,37)(21,33)(24,29)(28,39)(30,40)(32,34)(36,38);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,23)(13,29)(14,28)(15,30)(16,31)(18,20)(19,36)(21,34)(22,38)(24,25)(26,39)(27,40)(32,33)(35,37);(3,8)(5,7)(9,12)(10,18)(11,24)(13,28)(14,29)(15,30)(16,32)(17,20)(19,34)(21,36)(22,38)(23,25)(26,40)(27,39)(31,33)(35,37);(2,6)(5,7)(9,13,15,14)(10,19,21,20)(11,25,27,26)(12,28,30,29)(16,22,33,37)(17,18,34,36)(23,39,40,24)(31,35,32,38)Polycyclic presentation:
2^9|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, 6962120);