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 | #2407 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,16)(11,19,20,15)(12,23,22,17)(13,18,21,24)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,40,37);(1,2)(3,6)(4,7)(5,8)(9,11,14,20)(10,15,16,19)(12,21,22,13)(17,24,23,18)(25,27)(26,31)(28,37)(29,39)(30,36)(32,40)(33,38)(34,35);(1,3,5,4)(2,6,8,7)(10,16)(11,20)(12,22)(18,24)(27,36)(28,38)(31,35)(32,39);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,20)(21,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(9,12)(10,17)(11,21)(13,20)(14,22)(15,24)(16,23)(18,19)(25,28,30,38)(26,32,34,39)(27,37,36,33)(29,31,40,35);(9,13,14,21)(10,18,16,24)(11,22,20,12)(15,23,19,17)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,35,38,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7896813);