Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,36)(30,34)(31,32)(33,39)(35,38)(37,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(26,32)(27,34)(33,36)(35,40);(1,2)(3,4)(5,7)(6,8)(11,18)(14,15)(19,24)(22,23)(25,27,31,34)(26,29,32,30)(28,35,39,40)(33,37,36,38);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29,31,30)(26,27,32,34)(28,37,39,38)(33,35,36,40);(1,4,5,6)(2,3,7,8)(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,30,31,29)(26,34,32,27)(28,38,39,37)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635269);