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 | |
4443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,39)(38,40);(1,3)(2,6)(4,8)(5,7)(11,17)(15,21)(18,23)(22,24)(25,27)(26,32)(28,30)(29,38)(31,39)(33,40)(34,35)(36,37);(1,2)(3,4)(5,8)(6,7)(10,14)(15,21)(16,19)(22,24)(25,28,31,29)(26,33,36,34)(27,30,39,38)(32,40,37,35);(1,4)(2,7)(3,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,31,28)(26,34,36,33)(27,38,39,30)(32,35,37,40);(1,4)(2,3)(5,6)(7,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,30,31,38)(26,35,36,40)(27,28,39,29)(32,34,37,33);(1,2)(3,6)(4,7)(5,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,26)(27,32)(28,33)(29,34)(30,40)(31,36)(35,38)(37,39)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9625718);