Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| 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 | #60695 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,16)(12,25)(13,18)(14,20)(15,21)(17,24)(19,26)(22,31)(23,32)(27,40)(28,34)(29,36)(30,37)(33,39)(35,38);(1,2)(3,6)(4,8)(5,7)(9,11,15,24)(10,16,21,17)(12,26,23,14)(13,28,30,27)(18,34,37,40)(19,32,20,25)(22,38,39,29)(31,35,33,36);(9,12,14,11,15,23,26,24)(10,17,20,25,21,16,19,32)(13,27,29,22,30,28,38,39)(18,33,36,40,37,31,35,34);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,13,26,38,15,30,14,29)(10,18,19,35,21,37,20,36)(11,22,12,27,24,39,23,28)(16,31,25,40,17,33,32,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=fhi,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,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718328);