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 | #42299 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(26,31)(27,34)(28,36)(37,39);(1,2)(3,7)(4,5)(6,8)(10,15)(12,16)(17,22)(20,23)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(2,5)(6,7)(13,21)(17,22)(18,24)(20,23)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(3,8)(6,7)(9,11,14,19)(10,16,15,12)(13,18,21,24)(17,23,22,20)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3,4,8)(2,6,5,7)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8269392);