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 | |
4429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,16)(12,13)(14,17)(15,18)(19,24)(20,21)(22,23)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(1,3)(2,7)(4,5)(6,8)(11,18)(15,16)(19,21)(20,24)(25,26,31,28)(27,32,29,34)(30,33,38,36)(35,39,37,40);(1,2,6,4)(3,7,8,5)(9,11)(10,15)(12,19)(13,20)(14,18)(16,17)(21,22)(23,24)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(2,4)(3,8)(9,11)(10,16)(12,21)(13,20)(14,18)(15,17)(19,22)(23,24)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(1,4,6,2)(3,7,8,5)(9,12,14,22)(10,13,17,23)(11,19,18,21)(15,20,16,24)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,5,6,7)(2,3,4,8)(9,13)(10,12)(11,20)(14,23)(15,19)(16,21)(17,22)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227560);