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 | |
4424 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)(11,16)(12,21)(13,18)(14,22)(15,17)(19,23)(20,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(2,5)(6,7)(9,11,15,19)(10,16,17,23)(12,20,22,13)(14,18,21,24)(26,31)(32,33)(34,37)(39,40);(1,2)(3,6)(4,5)(7,8)(10,17)(11,19)(12,22)(18,24)(25,27,30,28)(26,32,31,33)(29,35,38,36)(34,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,20)(13,19)(15,22)(16,18)(17,21)(23,24)(25,28,30,27)(26,33,31,32)(29,36,38,35)(34,40,37,39);(9,13,15,20)(10,18,17,24)(11,12,19,22)(14,16,21,23)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,40)(36,39);(9,14)(10,12)(11,18)(13,23)(15,21)(16,20)(17,22)(19,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=ghi,e^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, 9830859);