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 | |
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:
(1,2)(3,4)(5,8)(6,7)(9,10,15,12)(11,13,16,19)(14,17,23,20)(18,21,22,24)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,37,39);(2,7)(3,5)(9,11,15,16)(10,13,12,19)(14,22,23,18)(17,24,20,21)(25,27,30,32)(26,31,34,28)(29,37,38,35)(33,36,40,39);(2,7)(4,8)(11,16)(13,19)(18,22)(21,24)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(1,3,6,5)(2,8,7,4)(9,12,15,10)(11,13,16,19)(14,20,23,17)(18,21,22,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,40)(37,38);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,12)(14,21)(15,19)(17,22)(18,20)(23,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,5,6,3)(2,4,7,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,23)(16,22)(19,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10268128);