Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| 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 | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10,15,17,13,19,14,20)(11,21,22,32,24,16,25,33)(12,26,30,36,28,34,29,18)(23,38,39,31,27,40,37,35);(1,3,5,7)(2,6,8,4)(9,11,13,24)(10,16,19,21)(12,27,28,23)(14,25,15,22)(17,33,20,32)(18,35,36,31)(26,38,34,40)(29,39,30,37);(1,4,5,6)(2,7,8,3)(10,17)(11,22)(14,15)(16,21)(18,34)(19,20)(23,37)(24,25)(26,36)(27,39)(29,30)(31,35);(2,8)(3,7)(9,12)(10,18)(11,23)(13,28)(14,30)(15,29)(16,31)(17,34)(19,36)(20,26)(21,35)(22,37)(24,27)(25,39)(32,40)(33,38);(11,24)(16,21)(22,25)(23,27)(31,35)(32,33)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411061);