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 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,21)(13,18)(14,15)(17,22)(19,20)(23,24);(2,3)(6,7)(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,22)(23,24)(25,26,31,35)(27,36,37,32)(28,29,33,39)(30,34,38,40);(1,3,5,2)(4,7,8,6)(10,15)(11,20)(17,21)(18,24)(25,27,31,37)(26,32,35,36)(28,38,33,30)(29,34,39,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,28,31,33)(26,29,35,39)(27,30,37,38)(32,40,36,34);(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,23)(20,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,39)(36,38)(37,40);(9,13,14,23)(10,18,15,24)(11,17,20,21)(12,16,22,19)(25,30)(26,34)(27,28)(29,36)(31,38)(32,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627582);