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 | #43362 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(6,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(26,32)(28,38)(33,39)(35,36);(1,2,5,4)(3,7,8,6)(11,17)(15,21)(18,23)(22,24)(25,26,31,32)(27,33,29,39)(28,34,38,37)(30,35,40,36);(1,3)(2,6)(4,7)(5,8)(10,14)(15,21)(16,19)(22,24)(25,27,31,29)(26,33,32,39)(28,36,38,35)(30,37,40,34);(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,34)(27,36)(29,35)(30,33)(31,38)(32,37)(39,40);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,27)(26,33,32,39)(28,35,38,36)(30,37,40,34);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,35)(27,37)(28,33)(29,34)(31,40)(32,36)(38,39)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770653);