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 | #50861 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,37)(39,40);(1,3,5,7)(2,6,8,4)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,26,31,36)(27,32,39,40)(28,33,37,38)(29,30,35,34);(1,4)(2,7)(3,8)(5,6)(10,15)(14,21)(16,19)(22,23)(25,27,31,39)(26,32,36,40)(28,29,37,35)(30,38,34,33);(1,3,5,7)(2,6,8,4)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28,31,37)(26,33,36,38)(27,29,39,35)(30,40,34,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29)(26,34)(27,37)(28,39)(30,36)(31,35)(32,33)(38,40);(1,2)(3,6)(4,7)(5,8)(9,12)(10,16)(11,17)(13,20)(14,22)(15,19)(18,24)(21,23)(25,30,31,34)(26,35,36,29)(27,38,39,33)(28,40,37,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9720134);