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 | #50866 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,24)(22,23)(25,26,31,35)(27,29,38,32)(28,39,37,33)(30,40,36,34);(1,2)(3,6)(4,7)(5,8)(11,20)(13,21)(16,19)(18,24)(25,27,31,38)(26,32,35,29)(28,30,37,36)(33,40,39,34);(1,3)(2,5)(4,8)(6,7)(9,11,15,20)(10,16,14,19)(12,21,23,13)(17,18,22,24)(25,28,31,37)(26,33,35,39)(27,36,38,30)(29,40,32,34);(3,8)(5,6)(9,12,15,23)(10,17,14,22)(11,13,20,21)(16,24,19,18)(25,29)(26,27)(28,40)(30,39)(31,32)(33,36)(34,37)(35,38);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,12)(14,24)(15,21)(16,22)(17,19)(20,23)(25,27)(26,29)(28,36)(30,37)(31,38)(32,35)(33,40)(34,39);(1,2)(3,6)(4,7)(5,8)(9,14,15,10)(11,16,20,19)(12,22,23,17)(13,24,21,18)(25,30,31,36)(26,34,35,40)(27,37,38,28)(29,39,32,33)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8811699);