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 | |
4461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,16,21,17)(12,18,23,13)(19,20,22,24)(25,26)(27,36)(28,29)(30,38)(31,32)(33,37)(34,40)(35,39);(1,2)(3,7)(4,5)(6,8)(10,15)(13,18)(16,17)(20,24)(26,32)(27,37)(29,34)(35,38);(2,5)(6,7)(9,11,14,21)(10,16,15,17)(12,22,23,19)(13,20,18,24)(25,27)(26,33)(28,38)(29,39)(30,34)(31,37)(32,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,19)(13,20)(14,21)(15,16)(18,24)(22,23)(25,28)(26,34)(27,38)(29,32)(30,33)(31,40)(35,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,19)(13,15)(14,23)(16,20)(17,24)(21,22)(25,29,31,34)(26,28,32,40)(27,39,37,30)(33,38,36,35);(1,3)(2,6)(4,8)(5,7)(9,13,14,18)(10,12,15,23)(11,20,21,24)(16,22,17,19)(25,30,31,39)(26,35,32,38)(27,34,37,29)(28,33,40,36)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8666211);