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 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,7)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,30,33)(27,31,35,28)(29,37,39,32)(34,36,40,38);(1,2,6,5)(3,4,8,7)(12,19)(16,22)(18,23)(20,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,3)(2,7)(4,5)(6,8)(10,15)(11,17)(16,22)(18,23)(25,27,30,35)(26,28,33,31)(29,34,39,40)(32,36,37,38);(1,4)(2,3)(5,8)(6,7)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,37)(39,40);(1,5,6,2)(3,4,8,7)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,40,37,34);(1,4)(2,8)(3,5)(6,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,32)(27,34)(28,36)(30,39)(31,38)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=ghi,e^2=h,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, 9854690);