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 | |
4818 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)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,30,31)(27,28,33,36)(29,37,40,32)(34,39,38,35);(1,3,4,7)(2,5,6,8)(10,14)(11,17)(16,21)(19,23)(25,27)(26,28)(29,38)(30,33)(31,36)(32,39)(34,40)(35,37);(1,2)(3,5)(4,6)(7,8)(12,20)(16,21)(19,23)(22,24)(26,31)(27,33)(32,37)(34,38);(1,3)(2,5)(4,7)(6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,27)(29,39)(30,36)(31,33)(32,38)(34,37)(35,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,29)(26,32)(27,34)(28,35)(30,40)(31,37)(33,38)(36,39);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313774);