Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,19)(11,21,24,16)(12,26,28,34)(14,17,15,20)(18,29,36,30)(22,33,25,32)(23,38,27,40)(31,39,35,37);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,27)(13,24)(14,25)(15,22)(17,33)(18,35)(19,21)(20,32)(23,28)(26,38)(29,39)(30,37)(31,36)(34,40);(10,17)(11,22)(14,15)(18,34)(19,20)(23,37)(24,25)(26,36)(27,39)(29,30)(32,33)(38,40);(2,6)(3,8)(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);(10,19)(16,21)(17,20)(18,36)(26,34)(31,35)(32,33)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,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, 7408335);