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 | |
4468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,20)(11,21,14,16)(12,18,23,17)(13,22,24,19)(25,26)(27,29)(28,32)(30,34)(31,35)(33,37)(36,40)(38,39);(3,7)(5,8)(9,11,15,14)(10,16,20,21)(12,22,23,19)(13,17,24,18)(25,27)(26,29)(28,36)(30,37)(31,38)(32,40)(33,34)(35,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,22)(13,21)(14,19)(15,23)(16,24)(18,20)(25,28)(26,32)(27,30)(29,34)(31,37)(33,35)(36,38)(39,40);(1,3,4,7)(2,5,6,8)(9,12)(10,18)(11,19)(13,21)(14,22)(15,23)(16,24)(17,20)(25,28)(26,33)(27,36)(29,34)(30,38)(31,37)(32,35)(39,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,24)(10,19,20,22)(11,17,14,18)(12,21,23,16)(25,29)(26,27)(28,34)(30,32)(31,39)(33,36)(35,38)(37,40);(9,14,15,11)(10,16,20,21)(12,19,23,22)(13,17,24,18)(25,30,31,36)(26,34,35,40)(27,37,38,28)(29,33,39,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8594780);