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 | |
4476 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)(4,5)(6,8)(9,10)(11,17)(12,19)(13,21)(14,15)(16,23)(18,20)(22,24)(25,26,30,31)(27,32,35,29)(28,36,39,33)(34,38,37,40);(2,5)(6,7)(9,11,14,19)(10,12,15,17)(13,22,23,18)(16,24,21,20)(26,31)(29,32)(33,36)(38,40);(1,2)(3,6)(4,5)(7,8)(10,15)(12,17)(16,21)(20,24)(25,27,30,35)(26,32,31,29)(28,34,39,37)(33,40,36,38);(1,3,4,8)(2,6,5,7)(9,12,14,17)(10,11,15,19)(13,20,23,24)(16,18,21,22)(25,27)(26,29)(28,37)(30,35)(31,32)(33,40)(34,39)(36,38);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28,30,39)(26,33,31,36)(27,34,35,37)(29,38,32,40);(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,29)(26,27)(28,38)(30,32)(31,35)(33,34)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8780822);