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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,17)(12,28)(13,19)(14,20)(15,21)(16,22)(18,30)(23,38)(24,34)(25,35)(26,36)(27,37)(29,40)(31,39)(32,33);(2,5)(6,8)(9,11)(10,17)(12,29)(13,31)(14,26)(15,25)(16,27)(18,38)(19,39)(20,36)(21,35)(22,37)(23,30)(24,32)(28,40)(33,34);(9,12)(10,18)(11,23)(13,27)(14,30)(15,31)(16,24)(17,33)(19,36)(20,34)(21,28)(22,39)(25,29)(26,32)(35,38)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,27)(14,32)(15,29)(16,23)(17,34)(18,36)(20,33)(21,40)(22,38)(25,31)(26,30)(28,37)(35,39);(9,14,16,15)(10,20,22,21)(11,25,27,26)(12,30,24,31)(13,32,23,29)(17,35,37,36)(18,34,39,28)(19,33,38,40)Polycyclic presentation:
2^9|a^2=f,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6346812);