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 | |
4321 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)(11,19)(12,16)(13,23)(14,18)(15,20)(17,21)(22,24)(28,36)(31,38)(34,40)(37,39);(1,2,5,8)(3,4,7,6)(9,11,14,20)(10,15,18,19)(12,21,22,13)(16,23,24,17)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,7)(5,8)(9,10,14,18)(11,15,20,19)(12,16,22,24)(13,17,21,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,20)(21,23)(22,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,39)(33,38)(35,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,22)(10,16,18,24)(11,13,20,21)(15,17,19,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,4,5,6)(2,7,8,3)(9,13)(10,17)(11,12)(14,21)(15,16)(18,23)(19,24)(20,22)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,e^2=g,f^2=gh,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, 9619664);