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 | |
4577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,15,20)(11,19,14,16)(12,13,18,17)(21,24,23,22)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2,5,3)(4,6,8,7)(9,11,15,14)(10,16,20,19)(12,21,18,23)(13,22,17,24)(26,27)(31,33)(32,34)(38,39);(2,3)(6,7)(12,18)(13,17)(21,23)(22,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,2,5,3)(4,6,8,7)(9,12,15,18)(10,17,20,13)(11,21,14,23)(16,24,19,22)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,2)(4,7,8,6)(9,13,15,17)(10,18,20,12)(11,22,14,24)(16,23,19,21)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,19,20,16)(12,23,18,21)(13,24,17,22)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038734);