Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
1959 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)(3,6)(4,8)(5,7)(9,10)(11,17)(12,15)(13,18)(14,19)(16,20)(21,25)(22,32)(23,33)(24,26)(27,30)(28,34)(29,35)(31,36)(37,39)(38,40);(1,3,6,2)(4,8,5,7)(9,11,16,25)(10,17,20,21)(12,26,18,32)(13,22,15,24)(14,29,31,38)(19,35,36,40)(23,28,39,30)(27,33,34,37);(9,12,15,20,16,18,13,10)(11,21,22,26,25,17,24,32)(14,27,30,36,31,34,28,19)(23,37,29,40,39,33,38,35);(1,4,6,5)(2,7,3,8)(9,13,16,15)(10,18,20,12)(11,22,25,24)(14,28,31,30)(17,32,21,26)(19,34,36,27)(23,29,39,38)(33,35,37,40);(1,5,6,4)(2,7,3,8)(9,14,13,28,16,31,15,30)(10,19,18,34,20,36,12,27)(11,23,24,38,25,39,22,29)(17,33,26,40,21,37,32,35)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=fi,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, 6718699);