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 | |
4445 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)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26)(27,35)(28,29)(30,36)(31,33)(32,38)(34,40)(37,39);(1,3,7,8)(2,4,6,5)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(25,27,31,38)(26,32,33,35)(28,34,39,36)(29,30,37,40);(1,2,7,6)(3,5,8,4)(10,17)(11,20)(12,22)(18,24)(26,33)(28,39)(32,35)(34,36);(1,4,7,5)(2,8,6,3)(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,28)(26,29)(27,36)(30,35)(31,39)(32,40)(33,37)(34,38);(1,5,7,4)(2,8,6,3)(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,29)(26,28)(27,30)(31,37)(32,34)(33,39)(35,36)(38,40);(1,6,7,2)(3,5,8,4)(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,30)(26,34)(27,37)(28,35)(29,38)(31,40)(32,39)(33,36)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,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, 9843000);