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 | |
4606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26)(27,32)(28,39)(29,34)(30,35)(31,36)(33,38)(37,40);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27,31,35)(26,32,36,30)(28,29,38,37)(33,40,39,34);(1,3)(2,4)(5,7)(6,8)(25,28,31,38)(26,33,36,39)(27,37,35,29)(30,40,32,34);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,34)(39,40);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,29,31,37)(26,34,36,40)(27,38,35,28)(30,39,32,33);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,30,31,32)(26,35,36,27)(28,40,38,34)(29,39,37,33)Polycyclic presentation:
2^9|a^2=g,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, 8365686);