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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,13,19,15,20,14,18)(11,21,23,16,25,38,24,32)(12,17,28,36,30,37,29,35)(22,33,39,31,27,40,26,34);(2,5)(3,7)(9,11,15,25)(10,16,20,32)(12,26,30,39)(13,24,14,23)(17,33,37,40)(18,38,19,21)(22,28,27,29)(31,36,34,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,21)(19,38)(20,32)(22,30)(26,28)(29,39)(31,37)(33,36)(35,40);(2,5)(6,8)(9,12,13,28,15,30,14,29)(10,17,19,36,20,37,18,35)(11,22,23,39,25,27,24,26)(16,31,38,40,32,34,21,33);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,32,38)(17,35,37,36)(22,39,27,26)(31,33,34,40)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=fi,e^2=h,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, 7053018);