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) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,19)(11,20,23,17)(12,24,14,15)(13,21,22,18)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11,16,23)(10,17,19,20)(12,21,14,18)(13,15,22,24)(27,36)(29,37)(32,35)(34,40);(1,3)(2,4)(5,6)(7,8)(9,12,16,14)(10,15,19,24)(11,21,23,18)(13,17,22,20)(25,27,31,36)(26,32,30,35)(28,37,39,29)(33,34,38,40);(3,6)(4,8)(9,13,16,22)(10,18,19,21)(11,15,23,24)(12,20,14,17)(25,28,31,39)(26,33,30,38)(27,29,36,37)(32,40,35,34);(1,4,5,8)(2,6,7,3)(9,14,16,12)(10,15,19,24)(11,21,23,18)(13,20,22,17)(25,29)(26,34)(27,28)(30,40)(31,37)(32,38)(33,35)(36,39);(1,2)(3,4)(5,7)(6,8)(9,15)(10,14)(11,22)(12,19)(13,23)(16,24)(17,18)(20,21)(25,30,31,26)(27,35,36,32)(28,38,39,33)(29,34,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068908);