Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1811 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)(9,10,14,18,15,20,13,19)(11,21,24,16,25,33,31,22)(12,17,29,35,30,27,28,36)(23,34,26,32,39,40,38,37);(1,2)(3,7)(4,6)(5,8)(9,11,14,24,15,25,13,31)(10,16,19,21,20,22,18,33)(12,26,28,23,30,38,29,39)(17,34,35,32,27,40,36,37);(1,3)(2,5)(4,8)(6,7)(9,10)(11,22)(12,27)(13,18)(14,19)(15,20)(16,25)(17,30)(21,31)(23,32)(24,33)(26,34)(28,36)(29,35)(37,39)(38,40);(3,8)(5,7)(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,32)(18,35)(19,36)(20,27)(21,34)(22,37)(24,26)(25,39)(31,38)(33,40);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,24,25,31)(12,28,30,29)(16,21,22,33)(17,35,27,36)(23,26,39,38)(32,34,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910269);