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 | |
4459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,17)(16,22)(23,24)(25,26)(27,31)(28,36)(29,30)(32,38)(33,35)(34,40)(37,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,3,5,6)(2,4,7,8)(26,30)(31,33)(32,36)(39,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,2)(3,4)(5,7)(6,8)(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,4)(2,6)(3,7)(5,8)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|c^2=hi,e^2=g,f^2=g,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, 10009745);