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 | #23956 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1894 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,14)(13,18)(15,17)(16,23)(19,31)(21,33)(22,37)(24,34)(25,28)(26,40)(27,29)(30,36)(32,38)(35,39);(1,3,5,4)(2,6,8,7)(9,11,14,23)(10,16,12,20)(13,26,28,39)(15,21,31,24)(17,34,19,33)(18,35,25,40)(22,29,38,30)(27,37,36,32);(10,17)(11,21)(12,19)(15,31)(16,20)(18,27)(22,26)(23,24)(25,36)(29,30)(32,37)(38,39);(1,3,5,4)(2,7,8,6)(9,12)(10,14)(11,16)(13,27)(15,17)(18,29)(19,31)(20,23)(21,33)(22,35)(24,34)(25,30)(26,32)(28,36)(37,39)(38,40);(1,4,5,3)(2,6,8,7)(9,13)(10,18)(11,22)(12,25)(14,28)(15,29)(16,32)(17,27)(19,36)(20,37)(21,26)(23,38)(24,39)(30,31)(33,40)(34,35)Polycyclic presentation:
2^9|b^2=f,d^2=fi,e^2=fi,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, 6485868);