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 | #47002 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(11,17)(12,20)(15,18)(16,21)(25,26,31,34)(27,33,29,32)(28,30,38,40)(35,36,37,39);(1,2,4,6)(3,5,7,8)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,27,31,29)(26,32,34,33)(28,37,38,35)(30,36,40,39);(1,3)(2,5)(4,7)(6,8)(10,14)(15,18)(16,21)(23,24)(25,27,31,29)(26,32,34,33)(28,35,38,37)(30,39,40,36);(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,28)(26,30)(27,35)(29,37)(31,38)(32,39)(33,36)(34,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29,31,27)(26,33,34,32)(28,35,38,37)(30,39,40,36);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,30)(26,28)(27,36)(29,39)(31,40)(32,37)(33,35)(34,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8839845);