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 | |
4748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,13)(12,15)(14,20)(16,18)(17,22)(19,23)(21,24)(25,26,31,34)(27,32,30,40)(28,33,38,29)(35,36,37,39);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,13)(14,19)(15,18)(17,23)(20,21)(22,24)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(2,7)(3,8)(10,12)(11,18)(17,20)(19,24)(25,28,31,38)(26,29,34,33)(27,35,30,37)(32,39,40,36);(1,3)(2,4)(5,8)(6,7)(9,12,15,10)(11,13,18,16)(14,20,22,17)(19,21,24,23)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,18)(14,21)(15,16)(17,19)(20,24)(22,23)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,4,5,6)(2,8,7,3)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9968145);