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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2058 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)(11,23)(12,27)(13,19)(14,20)(15,21)(16,24)(17,22)(18,29)(25,34)(26,33)(28,39)(30,35)(31,37)(32,38)(36,40);(2,5)(6,8)(9,11)(10,12)(13,25)(14,19)(15,27)(16,18)(17,35)(20,32)(21,23)(22,29)(24,30)(26,36)(28,40)(31,39)(33,37)(34,38);(9,12,15,23)(10,18,21,35)(11,24,27,22)(13,31,32,36)(14,28,34,33)(16,30,17,29)(19,25,38,20)(26,37,39,40);(1,3,4,7)(2,6,5,8)(9,13,15,32)(10,19,21,38)(11,25,27,20)(12,14,23,34)(16,33,17,28)(18,37,35,40)(22,39,24,26)(29,31,30,36);(9,14,15,34)(10,20,21,25)(11,26,27,39)(12,28,23,33)(13,30,32,29)(16,31,17,36)(18,19,35,38)(22,40,24,37)Polycyclic presentation:
2^9|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=ehi,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, 6482979);