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 | |
4201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,15)(12,19)(13,21)(14,17)(16,23)(18,20)(22,24)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(1,2,6,5)(3,8,7,4)(9,11,14,19)(10,15,17,12)(13,18,23,24)(16,22,21,20)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,15)(10,11,17,19)(13,22,23,20)(16,24,21,18)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,2,6,5)(3,4,7,8)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(1,4,6,8)(2,7,5,3)(9,10)(11,15)(12,19)(13,16)(14,17)(18,22)(20,24)(21,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,5,6,2)(3,8,7,4)(9,13,14,23)(10,16,17,21)(11,18,19,24)(12,20,15,22)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957576);