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 |
| prime radical | #44227 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,15)(11,17,19,12)(13,21,23,16)(18,20,22,24)(25,26,31,29)(27,32,37,39)(28,34,38,33)(30,40,36,35);(2,6)(4,7)(9,11,14,19)(10,12,15,17)(13,22,23,18)(16,24,21,20)(25,27,31,37)(26,32,29,39)(28,30,38,36)(33,35,34,40);(1,2)(3,4)(5,6)(7,8)(10,15)(12,17)(16,21)(20,24)(25,28,31,38)(26,33,29,34)(27,36,37,30)(32,40,39,35);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,34)(27,30)(29,33)(31,38)(32,40)(35,39)(36,37);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,29,31,26)(27,32,37,39)(28,33,38,34)(30,40,36,35);(1,3)(2,4)(5,8)(6,7)(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,30)(26,35)(27,28)(29,40)(31,36)(32,33)(34,39)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8717287);