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 | |
4326 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)(3,4)(5,8)(6,7)(9,10,14,17)(11,15,19,13)(12,16,22,24)(18,21,20,23)(27,33)(30,34)(35,39)(38,40);(1,3)(2,4)(5,6)(7,8)(9,11,14,19)(10,13,17,15)(12,20,22,18)(16,21,24,23)(25,26,29,32)(27,34,33,30)(28,31,37,36)(35,40,39,38);(1,2,5,8)(3,7,6,4)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,4)(2,6)(3,8)(5,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,21)(20,22)(23,24)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,18)(13,23)(14,22)(15,21)(17,24)(19,20)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,3)(2,7)(4,8)(5,6)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,20,24,18)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854516);