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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2049 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,19)(14,20)(15,21)(16,23)(17,26)(18,30)(22,37)(24,28)(25,29)(27,32)(31,40)(33,35)(34,39)(36,38);(2,5)(6,8)(9,11,15,26)(10,17,21,12)(13,24,33,20)(14,35,28,19)(16,27,37,18)(22,32,23,30)(25,31,39,36)(29,38,34,40);(9,12,15,17)(10,18,21,27)(11,22,26,23)(13,31,33,36)(14,29,28,34)(16,32,37,30)(19,24,35,20)(25,38,39,40);(1,3,4,7)(2,6,5,8)(9,13,15,33)(10,19,21,35)(11,24,26,20)(12,28,17,14)(16,34,37,29)(18,38,27,40)(22,25,23,39)(30,36,32,31);(9,14,15,28)(10,20,21,24)(11,25,26,39)(12,29,17,34)(13,32,33,30)(16,31,37,36)(18,19,27,35)(22,38,23,40)Polycyclic presentation:
2^9|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, 6482641);