Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
2054 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)(10,16)(11,20)(13,14)(18,32)(19,29)(22,37)(23,30)(24,36)(25,39)(26,27)(31,34)(38,40);(1,2)(3,7)(4,6)(5,8)(9,10)(11,21)(12,24)(13,29)(14,16)(15,19)(17,23)(18,27)(20,31)(22,38)(25,40)(26,36)(28,32)(30,34)(33,37)(35,39);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,25)(13,20)(14,30)(15,23)(16,31)(18,35)(19,21)(22,28)(24,38)(26,39)(27,37)(29,34)(32,40)(33,36);(3,8)(5,7)(9,12,15,28)(10,18,19,36)(11,22,23,25)(13,26,14,27)(16,32,29,24)(17,33,21,35)(20,37,30,39)(31,38,34,40);(2,6)(5,7)(9,13,15,14)(10,16,19,29)(11,20,23,30)(12,26,28,27)(17,31,21,34)(18,32,36,24)(22,37,25,39)(33,38,35,40)Polycyclic presentation:
2^9|c^2=hi,d^2=i,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,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, 6962110);