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 | |
4547 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,16,19)(13,17,22,23)(18,20,24,21)(25,26)(27,32)(28,38)(29,34)(30,31)(33,39)(35,40)(36,37);(2,5)(6,7)(10,15)(12,19)(17,23)(20,21)(25,27,31,37)(26,32,30,36)(28,35,39,29)(33,34,38,40);(1,2)(3,6)(4,5)(7,8)(25,28,31,39)(26,33,30,38)(27,29,37,35)(32,40,36,34);(1,3,4,8)(2,6,5,7)(9,11,14,16)(10,12,15,19)(13,18,22,24)(17,20,23,21)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,34)(36,40);(9,12)(10,16)(11,15)(13,21)(14,19)(17,18)(20,22)(23,24)(25,29,31,35)(26,34,30,40)(27,28,37,39)(32,38,36,33);(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,30,31,26)(27,36,37,32)(28,38,39,33)(29,34,35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820148);