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 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,5,4)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26)(27,36)(28,29)(30,40)(31,35)(32,38)(33,39)(34,37);(1,3,6,5)(2,7,8,4)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27)(26,32)(28,30)(29,34)(31,38)(33,40)(35,36)(37,39);(1,2)(3,4)(5,7)(6,8)(10,15)(14,21)(16,19)(22,23)(25,28,31,39)(26,33,35,29)(27,30,38,37)(32,40,36,34);(1,4)(2,5)(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28)(26,29)(27,37)(30,38)(31,39)(32,40)(33,35)(34,36);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29)(26,28)(27,34)(30,32)(31,33)(35,39)(36,37)(38,40);(1,5,6,3)(2,7,8,4)(9,12,13,20)(10,16,15,19)(11,17,18,24)(14,22,21,23)(25,30,31,37)(26,34,35,40)(27,28,38,39)(29,36,33,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,f^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, 9844917);