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 |
| prime radical | #43649 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4380 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)(11,12)(13,23)(14,19)(15,17)(16,21)(18,20)(22,24)(25,26,30,34)(27,31,37,32)(28,29,33,39)(35,36,40,38);(3,7)(5,8)(9,11,15,21)(10,16,17,12)(13,14,20,24)(18,19,23,22)(25,27)(26,31)(28,35)(29,36)(30,37)(32,34)(33,40)(38,39);(1,2)(3,5)(4,6)(7,8)(10,17)(12,16)(18,23)(19,22)(25,27,30,37)(26,31,34,32)(28,35,33,40)(29,36,39,38);(1,3,4,7)(2,5,6,8)(9,12,15,16)(10,11,17,21)(13,22,20,19)(14,23,24,18)(25,27)(26,32)(28,35)(29,38)(30,37)(31,34)(33,40)(36,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,14)(12,22)(15,20)(16,19)(17,23)(21,24)(25,28)(26,29)(27,35)(30,33)(31,36)(32,38)(34,39)(37,40);(9,14)(10,19)(11,20)(12,18)(13,21)(15,24)(16,23)(17,22)(25,29)(26,33)(27,36)(28,34)(30,39)(31,40)(32,35)(37,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=h,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, 8648837);