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 | |
4373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10)(11,15)(12,13)(14,17)(16,20)(18,19)(21,24)(22,23)(26,30)(31,36)(32,37)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,22)(23,24)(28,35)(32,37)(33,39)(38,40);(1,3,5,8)(2,7,6,4)(9,11,14,20)(10,15,17,16)(12,18,22,21)(13,19,23,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(3,8)(4,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,17)(18,22)(23,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,22)(10,13,17,23)(11,18,20,21)(15,19,16,24)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,13)(10,12)(11,19)(14,23)(15,18)(16,21)(17,22)(20,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40)Polycyclic presentation:
2^9|c^2=gi,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113944);