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 | #37713 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26)(27,32)(28,37)(29,33)(30,36)(31,35)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27,31,38)(26,32,35,39)(28,34,29,36)(30,37,40,33);(2,5)(6,8)(11,17)(14,21)(18,23)(22,24)(25,28,31,29)(26,33,35,37)(27,36,38,34)(30,32,40,39);(3,7)(6,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,29,31,28)(26,33,35,37)(27,34,38,36)(30,32,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,30)(26,34)(27,37)(28,39)(29,32)(31,40)(33,38)(35,36);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,26)(27,32)(28,33)(29,37)(30,34)(31,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8417126);