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 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(27,33)(30,37)(34,39)(38,40);(2,3)(6,7)(11,17)(14,21)(18,23)(22,24)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,3,5,2)(4,6,8,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,4,5,8)(2,6,3,7)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820836);