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 | |
4418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,16)(11,14,18,23)(12,15,21,19)(17,22,20,24)(27,34)(30,32)(35,40)(38,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,22)(18,23)(19,21)(20,24)(25,26)(27,32)(28,36)(29,33)(30,34)(31,37)(35,38)(39,40);(1,3,5,6)(2,4,7,8)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22)(25,28,29,37)(26,31,33,36)(27,35,34,40)(30,38,32,39);(1,2)(3,4)(5,7)(6,8)(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,24,19,22)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,38)(37,40);(1,4)(2,6)(3,7)(5,8)(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24)(25,26)(27,30)(28,36)(29,33)(31,37)(32,34)(35,39)(38,40)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8571939);