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 | #35365 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,18)(11,16,19,24)(12,22,20,14)(13,17,23,21)(25,26,30,31)(27,32,36,40)(28,29,33,39)(34,38,37,35);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,13)(14,21)(15,19)(17,22)(18,24)(20,23)(26,31)(28,33)(32,40)(34,37);(1,2)(3,5)(4,6)(7,8)(11,19)(12,20)(14,22)(16,24)(25,27,30,36)(26,32,31,40)(28,34,33,37)(29,38,39,35);(9,12)(10,14)(11,13)(15,20)(16,21)(17,24)(18,22)(19,23)(25,27)(26,32)(28,37)(29,35)(30,36)(31,40)(33,34)(38,39);(1,3)(2,5)(4,7)(6,8)(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,21)(22,24)(25,28,30,33)(26,29,31,39)(27,34,36,37)(32,38,40,35);(1,2)(3,5)(4,6)(7,8)(9,14)(10,12)(11,21)(13,16)(15,22)(17,19)(18,20)(23,24)(25,29)(26,33)(27,35)(28,31)(30,39)(32,34)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835849);