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 | #42505 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,18,16)(12,20,23,17)(19,24,21,22)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(1,2,5,4)(3,6,7,8)(10,15)(11,18)(17,20)(19,21)(25,27,31,28)(26,32,33,36)(29,37,39,34)(30,35,40,38);(2,4)(6,8)(10,15)(13,16)(17,20)(22,24)(25,28,31,27)(26,32,33,36)(29,37,39,34)(30,38,40,35);(3,7)(6,8)(9,11,14,18)(10,16,15,13)(12,21,23,19)(17,22,20,24)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(1,3,5,7)(2,6,4,8)(9,12,14,23)(10,17,15,20)(11,19,18,21)(13,22,16,24)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36);(1,4,5,2)(3,8,7,6)(9,13,14,16)(10,18,15,11)(12,22,23,24)(17,21,20,19)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8248935);