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 | #54516 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4161 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)(3,8)(4,7)(5,6)(11,12)(15,16)(19,20)(22,23)(25,26)(27,29)(28,36)(30,34)(31,32)(33,39)(35,38)(37,40);(2,6)(3,7)(9,10,14,18)(11,15,12,16)(13,17,21,24)(19,22,20,23)(26,32)(27,34)(33,36)(35,40);(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,31,34)(26,29,32,30)(28,35,39,40)(33,37,36,38);(2,6)(4,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(1,3,5,7)(2,4,6,8)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,29,31,30)(26,27,32,34)(28,37,39,38)(33,35,36,40);(1,4)(2,7)(3,6)(5,8)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,30,31,29)(26,34,32,27)(28,38,39,37)(33,40,36,35)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=i,e^2=gh,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, 8411998);