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 |
| prime radical | #54632 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4174 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)(3,5)(4,7)(6,8)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,26,31,32)(27,33,36,38)(28,37,35,34)(29,39,40,30);(2,3)(6,7)(12,20)(17,22)(19,21)(23,24)(26,32)(29,40)(33,38)(34,37);(1,2,5,3)(4,6,8,7)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,27,31,36)(26,33,32,38)(28,30,35,39)(29,34,40,37);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,28)(26,34)(27,30)(29,38)(31,35)(32,37)(33,40)(36,39);(1,4,5,8)(2,6,3,7)(9,13,15,11)(10,18,14,16)(12,21,20,19)(17,24,22,23)(25,29)(26,30)(27,34)(28,38)(31,40)(32,39)(33,35)(36,37);(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,30)(26,29)(27,35)(28,36)(31,39)(32,40)(33,34)(37,38)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8411968);