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 | |
4425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,4,5)(25,26)(27,32)(28,34)(29,30)(31,36)(33,38)(35,40)(37,39);(1,3)(2,5)(4,6)(7,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,31,37)(26,32,36,39)(28,29,38,35)(30,33,40,34);(1,3)(2,7)(4,6)(5,8)(10,14)(11,18)(12,21)(23,24)(25,28,31,38)(26,33,36,34)(27,35,37,29)(30,39,40,32);(2,8)(3,4)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,29,31,35)(26,30,36,40)(27,28,37,38)(32,34,39,33);(1,4,6,3)(2,5,8,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,28)(26,34)(27,29)(30,32)(31,38)(33,36)(35,37)(39,40);(1,5)(2,3)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30)(26,35)(27,33)(28,32)(29,36)(31,40)(34,37)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965810);