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:
(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,26,30,34)(27,35,36,31)(28,32,38,40)(29,33,37,39);(3,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,37)(35,36)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,17,15)(12,18,21,24)(16,22,20,23)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,40,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,28,30,38)(26,32,34,40)(27,29,36,37)(31,39,35,33);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,20)(13,17)(15,19)(16,21)(18,23)(22,24)(25,29)(26,33)(27,28)(30,37)(31,40)(32,35)(34,39)(36,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,37)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9089044);