Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | |
1908 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)(9,10,14,20,16,22,15,21)(11,23,27,17,28,40,26,30)(12,29,31,25,33,37,32,18)(13,19,36,39,35,24,34,38);(2,5)(3,7)(9,11)(10,17)(12,24)(13,29)(14,26)(15,27)(16,28)(18,36)(19,33)(20,23)(21,40)(22,30)(25,34)(31,39)(32,38)(35,37);(1,2)(3,6)(4,5)(7,8)(9,12,15,32,16,33,14,31)(10,18,20,29,22,25,21,37)(11,24,27,38,28,19,26,39)(13,30,34,40,35,17,36,23);(2,5)(6,8)(9,13,14,36,16,35,15,34)(10,19,20,39,22,24,21,38)(11,25,27,37,28,18,26,29)(12,30,31,23,33,17,32,40);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,31,33,32)(13,34,35,36)(17,23,30,40)(18,29,25,37)(19,38,24,39)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,e^2=hi,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,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063418);