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 | |
4103 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,20,22,16)(12,18,23,17)(13,24,21,19)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,13)(14,20)(15,22)(17,24)(18,19)(21,23)(27,33)(30,37)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12,15,23)(10,17,14,18)(11,13,22,21)(16,24,20,19)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,21)(13,22)(14,17)(15,23)(16,24)(19,20)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(9,13,15,21)(10,19,14,24)(11,12,22,23)(16,18,20,17)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(9,14,15,10)(11,20,22,16)(12,18,23,17)(13,19,21,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=gh,e^2=h,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, 8656899);