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 | |
4556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(3,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(26,32)(27,28)(34,37)(35,36);(1,2,5,4)(3,6,7,8)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,4,5,2)(3,8,7,6)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=ghi,f^2=i,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, 8366330);