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 | |
4682 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)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(26,32)(27,28)(30,34)(36,38);(1,2)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,31,32)(27,35,28,33)(29,34,40,30)(36,37,38,39);(2,5)(6,7)(10,14)(15,17)(16,20)(22,23)(25,27,31,28)(26,33,32,35)(29,36,40,38)(30,37,34,39);(2,5)(6,7)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,31,27)(26,33,32,35)(29,36,40,38)(30,39,34,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,34)(27,36)(28,38)(30,32)(31,40)(33,39)(35,37);(1,3)(2,6)(4,8)(5,7)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30,31,34)(26,29,32,40)(27,37,28,39)(33,36,35,38)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^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, 10230164);