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 | |
4470 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,19,20,16)(12,17,18,13)(21,22,24,23)(25,26,30,31)(27,28,34,35)(29,32,37,38)(33,36,39,40);(1,2,4,6)(3,5,8,7)(10,15)(11,20)(13,17)(21,24)(26,31)(28,35)(29,37)(33,39);(1,2)(3,5)(4,6)(7,8)(10,15)(12,18)(16,19)(21,24)(25,27,30,34)(26,28,31,35)(29,33,37,39)(32,36,38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,23)(18,24)(25,28,30,35)(26,27,31,34)(29,36,37,40)(32,33,38,39);(9,12,14,18)(10,17,15,13)(11,21,20,24)(16,23,19,22)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36)(37,39)(38,40);(9,13,14,17)(10,18,15,12)(11,22,20,23)(16,24,19,21)(25,29)(26,32)(27,33)(28,36)(30,37)(31,38)(34,39)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10256274);