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)(3,7)(4,5)(6,8)(10,15)(11,17)(18,23)(19,24)(25,26,31,32)(27,34,28,33)(29,35,39,38)(30,36,40,37);(2,4)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(26,32)(27,28)(35,38)(36,37);(1,3,5,8)(2,6,4,7)(25,27,31,28)(26,33,32,34)(29,37,39,36)(30,35,40,38);(1,4,5,2)(3,7,8,6)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,28,31,27)(26,34,32,33)(29,36,39,37)(30,38,40,35);(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,29)(26,35)(27,37)(28,36)(30,34)(31,39)(32,38)(33,40);(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,30)(26,36)(27,35)(28,38)(29,33)(31,40)(32,37)(34,39)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,e^2=gh,f^2=gh,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, 8004271);