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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10,14,19)(11,21,24,15)(12,26,29,36)(13,18,31,20)(16,28,35,30)(17,25,33,23)(22,38,40,34)(27,32,39,37);(1,3,5,7)(2,6,8,4)(9,11,14,24)(10,15,19,21)(12,27,29,39)(13,25,31,23)(16,34,35,38)(17,20,33,18)(22,28,40,30)(26,37,36,32);(1,4,5,6)(2,7,8,3)(9,12,14,29)(10,16,19,35)(11,22,24,40)(13,28,31,30)(15,32,21,37)(17,34,33,38)(18,26,20,36)(23,39,25,27);(2,8)(3,7)(9,11)(10,17)(12,22)(13,23)(14,24)(15,18)(16,34)(19,33)(20,21)(25,31)(26,32)(27,30)(28,39)(29,40)(35,38)(36,37);(9,13,14,31)(10,18,19,20)(11,23,24,25)(12,28,29,30)(15,33,21,17)(16,26,35,36)(22,39,40,27)(32,38,37,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,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, 7409253);