Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574075 |
| 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 | |
1750 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,19)(13,20)(14,30)(15,21)(16,22)(18,24)(23,32)(25,35)(26,33)(27,34)(28,37)(29,38)(31,36)(39,40);(2,6)(7,8)(9,11)(10,17)(12,16)(13,27)(14,24)(15,26)(18,30)(19,22)(20,34)(21,33)(23,31)(25,28)(29,39)(32,36)(35,37)(38,40);(9,11,14,24,16,12,15,26)(10,18,21,17,22,33,30,19)(13,23,28,27,29,25,31,39)(20,34,36,32,38,40,37,35);(1,3,5,4)(2,7,6,8)(9,12,16,11)(10,19,22,17)(13,27,29,39)(14,26,15,24)(18,30,33,21)(20,34,38,40)(23,28,25,31)(32,37,35,36);(1,4,5,3)(2,8,6,7)(9,13)(10,20)(11,23)(12,25)(14,28)(15,31)(16,29)(17,32)(18,34)(19,35)(21,36)(22,38)(24,27)(26,39)(30,37)(33,40)Polycyclic presentation:
2^9|c^2=g,d^2=fi,e^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=ehi,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, 6492256);