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 | |
4548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,16)(11,15,18,13)(12,20,21,17)(19,22,24,23)(25,26)(27,36)(28,29)(30,35)(31,33)(32,38)(34,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,18)(10,15,16,13)(12,19,21,24)(17,23,20,22)(25,27,31,38)(26,32,33,36)(28,35,34,40)(29,37,39,30);(1,3,5,4)(2,7,6,8)(10,16)(11,18)(12,21)(22,23)(26,33)(28,34)(32,36)(35,40);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,20)(18,24)(25,28,31,34)(26,29,33,39)(27,35,38,40)(30,32,37,36);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,21)(22,24)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,40)(35,36);(9,13,14,15)(10,11,16,18)(12,22,21,23)(17,19,20,24)(25,30,31,37)(26,35,33,40)(27,29,38,39)(28,32,34,36)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^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, 7882225);