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 | |
4471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,15)(11,12,17,16)(13,18,23,24)(19,20,21,22)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(2,3)(6,7)(10,15)(11,17)(18,24)(19,21)(27,36)(28,37)(32,38)(33,40);(1,2,5,3)(4,6,8,7)(25,27)(26,32)(28,29)(30,33)(31,36)(34,37)(35,38)(39,40);(1,3,5,2)(4,7,8,6)(9,11,14,17)(10,16,15,12)(13,21,23,19)(18,20,24,22)(25,28)(26,33)(27,29)(30,32)(31,37)(34,36)(35,40)(38,39);(1,4,5,8)(2,6,3,7)(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,24,21)(25,29)(26,30)(27,28)(31,34)(32,33)(35,39)(36,37)(38,40);(9,13,14,23)(10,18,15,24)(11,19,17,21)(12,20,16,22)(25,30)(26,34)(27,33)(28,38)(29,35)(31,39)(32,37)(36,40)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8599093);