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 | |
4388 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,31,28)(27,34,33,30)(29,32,37,36)(35,40,39,38);(2,5)(6,7)(10,14)(15,18)(16,21)(22,24)(29,37)(32,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(11,17)(12,20)(15,18)(16,21)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(1,3,4,8)(2,6,5,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,31,26)(27,34,33,30)(29,36,37,32)(35,40,39,38);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,29,31,37)(26,32,28,36)(27,35,33,39)(30,38,34,40);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,34)(26,33,28,27)(29,38,37,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835797);