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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,19,20,17)(12,22,21,18)(13,14,23,24)(25,26)(27,33)(28,38)(29,35)(30,36)(31,32)(34,39)(37,40);(1,2,5,4)(3,7,8,6)(10,16)(11,20)(13,23)(18,22)(25,27,32,29)(26,33,31,35)(28,30,39,37)(34,40,38,36);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,17,16,19)(12,13,21,23)(14,22,24,18)(25,28,32,39)(26,34,31,38)(27,37,29,30)(33,36,35,40);(3,8)(6,7)(9,12,15,21)(10,18,16,22)(11,13,20,23)(14,19,24,17)(25,29,32,27)(26,35,31,33)(28,37,39,30)(34,36,38,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,14)(11,21)(12,20)(15,23)(16,24)(17,22)(18,19)(25,30,32,37)(26,36,31,40)(27,28,29,39)(33,38,35,34);(1,4,5,2)(3,6,8,7)(9,14)(10,13)(11,22)(12,19)(15,24)(16,23)(17,21)(18,20)(25,31,32,26)(27,35,29,33)(28,34,39,38)(30,40,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8935795);