Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574063 |
| 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 | |
2078 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)(11,19)(12,17)(13,20)(14,21)(15,30)(16,22)(18,26)(23,35)(24,33)(25,32)(27,34)(28,37)(29,39)(31,38)(36,40);(2,5)(6,8)(9,11)(10,17)(12,14)(13,27)(15,24)(16,26)(18,30)(19,21)(20,36)(22,33)(23,29)(25,31)(28,40)(32,39)(34,37)(35,38);(9,12,15,26,14,11,16,24)(10,18,22,17,21,33,30,19)(13,25,29,27,28,23,31,40)(20,34,38,32,37,36,39,35);(1,3,4,7)(2,6,5,8)(9,11,14,12)(10,19,21,17)(13,27,28,40)(15,24,16,26)(18,30,33,22)(20,34,37,36)(23,31,25,29)(32,39,35,38);(9,13)(10,20)(11,23)(12,25)(14,28)(15,29)(16,31)(17,32)(18,34)(19,35)(21,37)(22,38)(24,40)(26,27)(30,39)(33,36)Polycyclic presentation:
2^9|c^2=g,d^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,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, 6490760);