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 | |
4511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,13)(14,15)(16,20)(18,23)(19,24)(21,22)(25,26)(27,32)(28,29)(30,39)(31,35)(33,38)(34,36)(37,40);(2,6)(3,7)(10,15)(12,21)(16,17)(18,24)(25,27,31,37)(26,32,35,40)(28,36,38,30)(29,34,33,39);(1,3)(2,4)(5,7)(6,8)(25,28,31,38)(26,33,35,29)(27,30,37,36)(32,34,40,39);(1,4)(2,3)(5,8)(6,7)(9,11,14,20)(10,16,15,17)(12,18,21,24)(13,19,22,23)(25,28)(26,29)(27,36)(30,37)(31,38)(32,34)(33,35)(39,40);(9,12,14,21)(10,13,15,22)(11,18,20,24)(16,19,17,23)(25,29)(26,28)(27,34)(30,40)(31,33)(32,36)(35,38)(37,39);(9,13)(10,12)(11,19)(14,22)(15,21)(16,18)(17,24)(20,23)(25,30,31,36)(26,34,35,39)(27,28,37,38)(29,32,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9770617);