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 | |
4247 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)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(1,2,5,4)(3,7,8,6)(9,11,14,19)(10,15,13,17)(12,18,21,24)(16,22,20,23)(27,33)(30,37)(34,39)(38,40);(1,3,5,8)(2,6,4,7)(12,21)(16,20)(18,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,5,2)(3,7,8,6)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23);(9,13,14,10)(11,15,19,17)(12,20,21,16)(18,22,24,23)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664254);