Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1931 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)(3,5)(4,6)(7,8)(9,10,14,21,16,22,15,20)(11,23,26,36,27,35,32,17)(12,28,30,18,25,38,31,39)(13,19,29,34,33,40,24,37);(1,2)(3,7)(4,6)(5,8)(9,11,16,27)(10,17,22,36)(12,29,25,24)(13,31,33,30)(14,32,15,26)(18,37,39,34)(19,38,40,28)(20,23,21,35);(1,3,4,8)(2,5,6,7)(9,12,16,25)(10,18,22,39)(11,24,27,29)(13,32,33,26)(14,30,15,31)(17,34,36,37)(19,23,40,35)(20,28,21,38);(3,8)(5,7)(9,13,15,24,16,33,14,29)(10,19,20,37,22,40,21,34)(11,25,32,30,27,12,26,31)(17,18,35,28,36,39,23,38);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,26,27,32)(12,30,25,31)(13,29,33,24)(17,35,36,23)(18,28,39,38)(19,37,40,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6963120);