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 | |
4627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,12,17,16)(13,21,22,18)(19,23,24,20)(26,30)(31,34)(32,36)(38,40);(1,2)(3,7)(4,6)(5,8)(10,15)(11,17)(13,22)(20,23)(25,26,29,30)(27,31,33,34)(28,36,37,32)(35,40,39,38);(1,2,4,6)(3,5,8,7)(10,15)(12,16)(18,21)(20,23)(27,33)(31,34)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,d^2=hi,e^2=g,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, 10268316);