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 | |
4535 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)(11,19,20,16)(12,21,22,17)(13,18,23,24)(25,26,31,30)(27,35,29,32)(28,33,38,34)(36,40,37,39);(1,2)(3,6)(4,7)(5,8)(10,15)(11,20)(17,21)(18,24)(25,27,31,29)(26,32,30,35)(28,36,38,37)(33,39,34,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,20)(17,21)(18,24)(25,28,31,38)(26,33,30,34)(27,36,29,37)(32,39,35,40);(3,8)(5,6)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,17,23,21)(25,28)(26,34)(27,36)(29,37)(30,33)(31,38)(32,40)(35,39);(1,2)(3,5)(4,7)(6,8)(9,12,14,22)(10,17,15,21)(11,18,20,24)(13,16,23,19)(25,29,31,27)(26,32,30,35)(28,37,38,36)(33,39,34,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,23)(10,18,15,24)(11,21,20,17)(12,19,22,16)(25,30,31,26)(27,35,29,32)(28,34,38,33)(36,40,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8609853);