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 |
| prime radical | #60580 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1886 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)(11,16)(12,15)(13,18)(14,17)(19,31)(20,34)(21,32)(22,33)(23,24)(25,30)(26,35)(27,28)(29,36)(37,39)(38,40);(1,3,5,4)(2,6,8,7)(9,11,15,23)(10,16,12,24)(13,26,30,38)(14,20,31,22)(17,34,19,33)(18,35,25,40)(21,28,37,29)(27,39,36,32);(10,17)(11,20)(12,19)(14,31)(16,24)(18,27)(21,26)(22,23)(25,36)(28,29)(32,39)(37,38);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,16)(13,27)(14,17)(18,28)(19,31)(20,33)(21,35)(22,34)(23,24)(25,29)(26,32)(30,36)(37,40)(38,39);(1,4,5,3)(2,6,8,7)(9,13)(10,18)(11,21)(12,25)(14,28)(15,30)(16,32)(17,27)(19,36)(20,26)(22,38)(23,37)(24,39)(29,31)(33,40)(34,35)Polycyclic presentation:
2^9|b^2=fi,d^2=f,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, 6485863);