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 | #41717 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(26,30)(28,35)(31,36)(34,39);(1,3,5,8)(2,6,7,4)(10,14)(15,17)(16,20)(23,24)(27,33)(31,36)(34,39)(38,40);(1,2)(3,4)(5,7)(6,8)(11,18)(15,17)(19,21)(23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4)(2,3)(5,6)(7,8)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23);(1,3)(2,4)(5,8)(6,7)(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4)(2,3)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8780941);