Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 | |
2006 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)(3,6)(4,8)(5,7)(9,10)(11,23)(12,29)(13,22)(14,20)(15,21)(16,19)(17,27)(18,26)(24,40)(25,35)(28,37)(30,34)(31,39)(32,38)(33,36);(1,3,6,2)(4,8,5,7)(9,11,15,27)(10,17,21,23)(12,30,31,14)(13,25,16,28)(18,38,36,40)(19,35,22,37)(20,39,34,29)(24,33,32,26);(9,12,13,26,15,31,16,33)(10,18,22,39,21,36,19,29)(11,24,28,30,27,32,25,14)(17,34,35,38,23,20,37,40);(1,4,6,5)(2,7,3,8)(9,13,15,16)(10,19,21,22)(11,25,27,28)(12,26,31,33)(14,32,30,24)(17,35,23,37)(18,29,36,39)(20,40,34,38);(1,5,6,4)(2,7,3,8)(9,14,13,24,15,30,16,32)(10,20,22,40,21,34,19,38)(11,26,28,31,27,33,25,12)(17,36,35,29,23,18,37,39)Polycyclic presentation:
2^9|b^2=f,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=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718823);