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 | #38505 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,20,21,16)(12,22,19,14)(13,18,23,24)(25,26,31,34)(27,35,37,32)(28,38,36,29)(30,33,39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,13)(14,18)(15,21)(17,20)(19,23)(22,24)(25,27,31,37)(26,32,34,35)(28,30,36,39)(29,33,38,40);(1,3,5,6)(2,4,7,8)(10,17)(11,21)(12,19)(18,24);(1,4)(2,6)(3,7)(5,8)(9,12,15,19)(10,14,17,22)(11,13,21,23)(16,18,20,24)(25,28,31,36)(26,29,34,38)(27,30,37,39)(32,33,35,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,18)(11,19)(12,21)(14,20)(15,23)(16,22)(17,24)(25,29)(26,28)(27,33)(30,32)(31,38)(34,36)(35,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,19)(11,18)(12,17)(13,20)(15,22)(16,23)(21,24)(25,30)(26,33)(27,36)(28,37)(29,35)(31,39)(32,38)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^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, 10009828);