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 | |
4586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(18,23)(19,24)(25,26,31,30)(27,34,32,28)(29,37,39,33)(35,36,38,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,27)(26,28)(29,38)(30,34)(31,32)(33,40)(35,39)(36,37);(1,3)(2,5)(4,8)(6,7)(25,28)(26,32)(27,30)(29,36)(31,34)(33,38)(35,37)(39,40);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,29,31,39)(26,33,30,37)(27,35,32,38)(28,36,34,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,38)(39,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,30,31,26)(27,34,32,28)(29,37,39,33)(35,40,38,36)Polycyclic presentation:
2^9|a^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003221);