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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24);(1,2,5,8)(3,6,4,7)(10,15)(11,18)(17,24)(19,21)(25,26,31,30)(27,35,37,32)(28,38,36,33)(29,34,39,40);(1,3,5,4)(2,6,8,7)(25,27,31,37)(26,32,30,35)(28,29,36,39)(33,34,38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,28,31,36)(26,33,30,38)(27,29,37,39)(32,34,35,40);(1,4,5,3)(2,7,8,6)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,29)(26,34)(27,36)(28,37)(30,40)(31,39)(32,38)(33,35);(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,30,31,26)(27,35,37,32)(28,33,36,38)(29,34,39,40)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608877);