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 | #37484 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,21)(16,19)(22,23)(25,26,31,34)(27,35,37,30)(28,38,40,32)(29,36,39,33);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,27)(26,30)(28,36)(29,38)(31,37)(32,39)(33,40)(34,35);(1,3)(2,5)(4,8)(6,7)(25,28,31,40)(26,32,34,38)(27,36,37,33)(29,30,39,35);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,29)(26,33)(27,32)(28,30)(31,39)(34,36)(35,40)(37,38);(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,26)(27,30)(28,32)(29,33)(31,34)(35,37)(36,39)(38,40);(1,2)(3,5)(4,7)(6,8)(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30)(26,27)(28,39)(29,40)(31,35)(32,36)(33,38)(34,37)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329392);