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 | |
4202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,14)(11,20,22,16)(12,18,23,17)(13,24,21,19)(26,30)(28,35)(31,36)(34,39);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,13)(14,20)(15,22)(17,24)(18,19)(21,23)(27,33)(31,36)(34,39)(38,40);(1,3,5,8)(2,6,4,7)(9,12,15,23)(10,17,14,18)(11,13,22,21)(16,24,20,19)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,21)(13,22)(14,17)(15,23)(16,24)(19,20)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,13,15,21)(10,19,14,24)(11,12,22,23)(16,18,20,17)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,14,15,10)(11,20,22,16)(12,18,23,17)(13,19,21,24)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,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, 8655111);