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 |
| prime radical | #2408 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,20,22,16)(12,21,24,17)(13,14,18,23)(25,26)(27,32)(28,34)(29,37)(30,35)(31,36)(33,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,22)(10,16,19,20)(12,23,24,14)(13,21,18,17)(25,27,31,39)(26,32,36,40)(28,30,38,37)(29,34,35,33);(1,3,5,6)(2,4,7,8)(25,28)(26,33)(27,37)(29,40)(30,39)(31,38)(32,35)(34,36);(1,4,5,8)(2,3,7,6)(9,12,15,24)(10,17,19,21)(11,14,22,23)(13,16,18,20)(25,28)(26,34)(27,30)(29,40)(31,38)(32,35)(33,36)(37,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,14)(11,21)(12,16)(15,18)(17,22)(19,23)(20,24)(25,29)(26,30)(27,34)(28,40)(31,35)(32,38)(33,39)(36,37);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,12)(13,19)(15,23)(16,17)(20,21)(22,24)(25,30)(26,35)(27,38)(28,39)(29,36)(31,37)(32,33)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8604726);