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 | #46040 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4352 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26,31,34)(27,35,37,32)(28,38,36,33)(29,39,40,30);(2,3)(6,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,31,37)(26,32,34,35)(28,30,36,39)(29,38,40,33);(1,2,5,3)(4,6,8,7)(25,28,31,36)(26,33,34,38)(27,30,37,39)(29,35,40,32);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,29,31,40)(26,30,34,39)(27,33,37,38)(28,35,36,32);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30)(26,29)(27,36)(28,37)(31,39)(32,38)(33,35)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352848);