Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437602 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,17)(12,25)(13,19)(14,31)(15,21)(16,20)(18,27)(22,38)(23,36)(24,34)(26,33)(28,37)(29,40)(30,39)(32,35);(1,3,5,4)(2,6,7,8)(9,11,16,24)(10,17,20,34)(12,26,29,30)(13,28,22,27)(14,23,15,32)(18,19,37,38)(21,35,31,36)(25,33,40,39);(9,12,16,29)(10,18,20,37)(11,22,24,13)(14,28,15,27)(17,33,34,39)(19,36,38,35)(21,40,31,25)(23,30,32,26);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,12)(14,30)(15,26)(16,22)(17,25)(18,36)(20,38)(21,33)(23,27)(24,29)(28,32)(31,39)(34,40)(35,37);(10,20)(17,34)(18,37)(19,38)(21,31)(25,40)(33,39)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346932);