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 | |
4557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,18)(11,19,20,15)(12,16,23,22)(13,17,24,21)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,18,19)(12,21,23,17)(13,16,24,22)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38);(11,20)(12,23)(15,19)(16,22)(25,27,29,35)(26,31,33,30)(28,37,38,34)(32,39,36,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,20)(21,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(9,12)(10,16)(11,21)(13,19)(14,23)(15,24)(17,20)(18,22)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,3,4,7)(2,5,6,8)(9,13,14,24)(10,17,18,21)(11,22,20,16)(12,15,23,19)(25,26)(27,31)(28,36)(29,33)(30,35)(32,38)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,f^2=g,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, 7876863);