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 | |
4393 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)(11,16)(12,14)(13,17)(15,19)(18,23)(20,21)(22,24)(25,26,31,36)(27,33,28,32)(29,34,39,37)(30,38,40,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27,31,28)(26,32,36,33)(29,38,39,35)(30,37,40,34);(2,5)(6,8)(13,21)(17,20)(18,24)(22,23)(25,27,31,28)(26,32,36,33)(29,35,39,38)(30,34,40,37);(3,7)(6,8)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,28,31,27)(26,33,36,32)(29,35,39,38)(30,34,40,37);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,29,31,39)(26,34,36,37)(27,35,28,38)(30,32,40,33);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30)(26,35)(27,37)(28,34)(29,32)(31,40)(33,39)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806498);