Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 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)(9,10,14,19)(11,21,24,34)(12,26,29,33)(13,18,15,20)(16,23,32,25)(17,30,27,28)(22,36,39,31)(35,38,37,40);(1,3,5,7)(2,6,8,4)(9,10)(11,21)(12,27)(13,20)(14,19)(15,18)(16,23)(17,29)(22,37)(24,34)(25,32)(26,30)(28,33)(31,40)(35,39)(36,38);(1,4,5,6)(2,7,8,3)(9,11,14,24)(10,16,19,32)(12,22,29,39)(13,23,15,25)(17,31,27,36)(18,21,20,34)(26,37,33,35)(28,38,30,40);(2,8)(3,7)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,33)(19,27)(20,26)(21,35)(23,38)(24,39)(25,40)(32,36)(34,37);(9,13,14,15)(10,18,19,20)(11,23,24,25)(12,28,29,30)(16,21,32,34)(17,33,27,26)(22,38,39,40)(31,35,36,37)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,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, 7409254);