Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | #24461 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,15,14)(11,22,26,18)(12,13,20,32)(16,33,17,21)(19,31,29,30)(23,37,40,39)(24,28,38,35)(25,27,36,34);(2,5)(6,8)(9,11,15,26)(10,18,14,22)(12,28,20,35)(13,24,32,38)(16,27,17,34)(19,37,29,39)(21,36,33,25)(23,30,40,31);(9,12,15,20)(10,19,14,29)(11,23,26,40)(13,21,32,33)(16,31,17,30)(18,24,22,38)(25,39,36,37)(27,35,34,28);(1,3,4,7)(2,6,5,8)(9,13,15,32)(10,20,14,12)(11,24,26,38)(16,29,17,19)(18,35,22,28)(21,31,33,30)(23,25,40,36)(27,39,34,37);(9,14,15,10)(11,25,26,36)(12,29,20,19)(13,30,32,31)(16,33,17,21)(18,34,22,27)(23,39,40,37)(24,28,38,35)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,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, 6483289);