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 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,15,13)(11,18,14,16)(12,21,22,17)(19,23,20,24)(25,26,31,30)(27,32,38,37)(28,34,29,33)(35,40,36,39);(1,2,6,5)(3,8,4,7)(9,11,15,14)(10,16,13,18)(12,19,22,20)(17,23,21,24)(25,27,31,38)(26,32,30,37)(28,35,29,36)(33,39,34,40);(1,3,6,4)(2,7,5,8)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,28,31,29)(26,33,30,34)(27,35,38,36)(32,39,37,40);(1,4,6,3)(2,8,5,7)(9,12)(10,17)(11,20)(13,21)(14,19)(15,22)(16,24)(18,23)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,5,6,2)(3,8,4,7)(9,13,15,10)(11,16,14,18)(12,21,22,17)(19,23,20,24)(25,29,31,28)(26,34,30,33)(27,36,38,35)(32,40,37,39);(9,14,15,11)(10,16,13,18)(12,20,22,19)(17,23,21,24)(25,30,31,26)(27,37,38,32)(28,34,29,33)(35,40,36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,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, 8412836);