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 | |
4515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(11,17)(12,20)(15,18)(16,21)(26,29)(32,34)(33,37)(39,40);(3,7)(5,8)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,26)(27,32)(28,34)(29,31)(30,33)(35,39)(36,40)(37,38);(1,2)(3,5)(4,6)(7,8)(10,14)(15,18)(16,21)(23,24)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839803);