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 | #2813 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(26,30)(27,33)(28,36)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(2,6)(4,7)(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,4,5,7)(2,8,6,3)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657003);