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 | |
4476 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)(3,4)(5,6)(7,8)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,23)(22,24)(25,26,31,28)(27,32,30,35)(29,33,37,36)(34,39,38,40);(2,6)(3,8)(10,15)(11,20)(17,24)(18,21)(25,27,31,30)(26,32,28,35)(29,34,37,38)(33,39,36,40);(3,8)(4,7)(10,15)(11,20)(12,22)(13,23)(26,28)(27,30)(29,37)(39,40);(1,3,5,8)(2,7,6,4)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,22)(23,24)(25,28,31,26)(27,32,30,35)(29,33,37,36)(34,40,38,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,21)(13,19)(14,22)(15,24)(16,23)(18,20)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,23)(10,18,15,21)(11,17,20,24)(12,19,22,16)(25,30,31,27)(26,32,28,35)(29,38,37,34)(33,39,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,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, 8969320);