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 | |
4367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(10,15)(11,18)(17,24)(19,20)(25,26,31,34)(27,32,30,40)(28,33,38,29)(35,36,37,39);(3,8)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,23)(22,24)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(1,3)(2,6)(4,7)(5,8)(25,28,31,38)(26,29,34,33)(27,35,30,37)(32,39,40,36);(3,8)(6,7)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,23,24,21)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,19,24,20)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073211);