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 | #55253 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,12)(11,18,17,14)(13,16,22,20)(19,24,23,21)(25,26,31,32)(27,37,39,33)(28,36,30,34)(29,35,40,38);(1,2)(3,7)(4,5)(6,8)(26,32)(28,30)(33,37)(35,38);(2,5)(6,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27,31,39)(26,33,32,37)(28,35,30,38)(29,34,40,36);(2,5)(6,7)(9,12,15,10)(11,18,17,14)(13,16,22,20)(19,21,23,24)(25,28,31,30)(26,34,32,36)(27,35,39,38)(29,33,40,37);(1,2)(3,6)(4,5)(7,8)(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,29)(26,35)(27,36)(28,33)(30,37)(31,40)(32,38)(34,39);(1,3)(2,6)(4,8)(5,7)(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30,31,28)(26,36,32,34)(27,38,39,35)(29,37,40,33)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231249);