Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | #61179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1929 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,18)(13,19)(14,16)(15,20)(21,30)(22,32)(23,31)(24,27)(25,34)(26,35)(28,33)(29,36)(37,39)(38,40);(1,3,5,4)(2,6,8,7)(9,10,13,19)(11,17,23,31)(12,24,26,36)(14,20,15,16)(18,29,35,27)(21,33,28,30)(22,37,38,34)(25,40,39,32);(1,3,5,4)(2,7,8,6)(10,16)(11,21)(14,15)(17,31)(18,24)(19,20)(22,25)(23,28)(27,29)(32,40)(35,36)(38,39);(1,4,5,3)(2,6,8,7)(9,11,13,23)(10,17,19,31)(12,25,26,39)(14,21,15,28)(16,30,20,33)(18,34,35,37)(22,27,38,29)(24,40,36,32);(9,12)(10,18)(11,22)(13,26)(14,27)(15,29)(16,24)(17,32)(19,35)(20,36)(21,25)(23,38)(28,39)(30,34)(31,40)(33,37)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484794);