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 | |
4590 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)(10,14)(11,17)(16,19)(18,23)(25,26)(27,32)(28,39)(29,34)(30,35)(31,36)(33,38)(37,40);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,31,35)(26,32,36,30)(28,29,38,37)(33,40,39,34);(1,2)(3,6)(4,5)(7,8)(25,28,31,38)(26,33,36,39)(27,37,35,29)(30,40,32,34);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,34)(39,40);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29,31,37)(26,34,36,40)(27,38,35,28)(30,39,32,33);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,30,31,32)(26,35,36,27)(28,40,38,34)(29,39,37,33)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8365685);