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 | |
4353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,15)(11,13)(12,20)(23,24)(25,26,31,30)(27,34,32,28)(29,37,38,33)(35,36,39,40);(3,7)(6,8)(9,10)(11,16)(12,21)(13,18)(14,15)(17,20)(19,24)(22,23)(25,27,31,32)(26,28,30,34)(29,35,38,39)(33,36,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,16,15,18)(12,19,20,22)(17,23,21,24)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,39,37,35);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,22)(14,20)(15,21)(16,23)(18,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,18,15,16)(12,19,20,22)(17,23,21,24)(25,29,31,38)(26,33,30,37)(27,35,32,39)(28,36,34,40);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,30,31,26)(27,34,32,28)(29,37,38,33)(35,40,39,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8059367);