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 | |
4343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,21)(22,24)(25,26,30,34)(27,28,32,31)(29,37,38,33)(35,39,40,36);(1,2)(3,7)(4,5)(6,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23);(2,5)(6,7)(13,21)(16,20)(18,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(2,5)(6,7)(9,12,14,10)(11,15,19,17)(13,16,21,20)(18,23,24,22)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,36)(30,31)(33,40)(35,37)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,17)(12,15)(13,18)(14,19)(16,23)(20,22)(21,24)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=g,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, 10229743);