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 | |
4362 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)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,26,31,29)(27,34,33,30)(28,36,38,32)(35,37,39,40);(1,2)(3,8)(4,5)(6,7)(10,14)(15,22)(16,19)(23,24)(26,29)(27,33)(32,36)(35,39);(2,7)(3,4)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,40)(38,39);(1,3,6,4)(2,8,7,5)(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,4,6,3)(2,5,7,8)(9,11)(10,15)(12,17)(13,18)(14,22)(16,23)(19,24)(20,21)(25,29,31,26)(27,34,33,30)(28,36,38,32)(35,40,39,37);(1,5)(2,3)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,30,31,34)(26,33,29,27)(28,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8706345);