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 | #44013 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,16)(11,15,20,17)(12,13,19,18)(21,23,24,22)(25,26)(27,37)(28,34)(29,40)(30,36)(31,33)(32,39)(35,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,23)(14,20)(16,17)(18,22)(19,24)(25,27)(26,32)(28,35)(29,34)(30,38)(31,39)(33,37)(36,40);(1,3,5,4)(2,7,6,8)(10,16)(11,20)(12,19)(22,23)(26,33)(28,36)(32,37)(35,40);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,23)(19,24)(25,28,31,36)(26,34,33,30)(27,35,39,40)(29,37,38,32);(1,4,5,3)(2,8,6,7)(9,12)(10,18)(11,21)(13,16)(14,19)(15,22)(17,23)(20,24)(25,29,31,38)(26,35,33,40)(27,34,39,30)(28,37,36,32);(9,13,14,18)(10,19,16,12)(11,22,20,23)(15,21,17,24)(25,30,31,34)(26,36,33,28)(27,38,39,29)(32,40,37,35)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=g,e^2=g,f^2=h,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, 7882190);