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 | |
4473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,17,16)(13,18,22,23)(19,20,24,21)(25,26,32,37)(27,35,29,33)(28,34,39,38)(30,31,36,40);(2,4)(6,7)(10,15)(11,17)(18,23)(19,24)(25,27,32,29)(26,33,37,35)(28,36,39,30)(31,38,40,34);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,28,32,39)(26,34,37,38)(27,36,29,30)(31,35,40,33);(1,4,5,2)(3,7,8,6)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,29,32,27)(26,35,37,33)(28,30,39,36)(31,34,40,38);(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,30)(26,31)(27,28)(29,39)(32,36)(33,38)(34,35)(37,40);(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,31)(26,36)(27,38)(28,35)(29,34)(30,37)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,e^2=i,f^2=i,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, 8004481);