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 | |
4726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,18,16,12)(13,17,22,23)(19,24,21,20)(25,26)(27,36)(28,30)(29,39)(31,34)(32,35)(33,38)(37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(17,23)(19,21)(25,27)(26,33)(28,37)(29,34)(30,31)(32,38)(35,36)(39,40);(2,5)(6,8)(10,15)(12,18)(17,23)(20,24)(25,28,32,29)(26,30,35,39)(27,37,38,34)(31,36,40,33);(3,7)(6,8)(9,11,14,16)(10,12,15,18)(13,21,22,19)(17,24,23,20)(25,29,32,28)(26,30,35,39)(27,34,38,37)(31,36,40,33);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,21)(19,23)(22,24)(25,30)(26,28)(27,31)(29,35)(32,39)(33,37)(34,36)(38,40);(1,3,4,7)(2,6,5,8)(9,13,14,22)(10,17,15,23)(11,19,16,21)(12,20,18,24)(25,31)(26,34)(27,30)(28,36)(29,33)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8450977);