Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814811 |
| 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 | #60772 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 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)(11,20)(12,16)(13,18)(14,19)(15,24)(17,28)(21,22)(23,33)(25,26)(27,32)(29,36)(30,38)(31,39)(34,35)(37,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,21)(12,25)(13,17)(14,19)(15,24)(16,26)(18,28)(20,22)(23,37)(27,31)(29,36)(30,35)(32,39)(33,40)(34,38);(9,11,14,22)(10,15)(12,23,27,35)(13,24,28,36)(16,30)(17,20)(18,21)(19,29)(25,34)(26,38,39,40)(31,33)(32,37);(2,6)(5,7)(11,22)(15,29)(20,21)(23,35)(24,36)(30,37)(33,34)(38,40);(3,8)(5,7)(9,12,13,26,14,27,28,39)(10,16,18,25,19,32,17,31)(11,23,24,38,22,35,36,40)(15,30,21,34,29,37,20,33)Polycyclic presentation:
2^9|c^2=gh,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722007);