Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | |
1890 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,27)(13,20)(14,21)(15,33)(16,22)(17,25)(18,31)(23,37)(24,38)(26,35)(28,34)(29,36)(30,39)(32,40);(1,3,5,4)(2,6,8,7)(9,11,14,25)(10,17,21,19)(12,13,30,28)(15,26,16,29)(18,37,38,40)(20,27,34,39)(22,35,33,36)(23,31,32,24);(1,3,5,4)(2,7,8,6)(9,12)(10,18)(11,23)(13,29)(14,30)(15,24)(16,31)(17,34)(19,20)(21,38)(22,39)(25,32)(26,28)(27,33)(35,37)(36,40);(1,3,5,4)(2,7,8,6)(9,11)(10,19)(12,28)(13,30)(14,25)(15,26)(16,29)(17,21)(18,37)(20,39)(22,36)(23,31)(24,32)(27,34)(33,35)(38,40);(1,4,5,3)(2,6,8,7)(9,13,14,28)(10,20,21,34)(11,24,25,31)(12,29,30,26)(15,32,16,23)(17,18,19,38)(22,37,33,40)(27,36,39,35)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486234);