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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4271 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,20,23,24)(25,26,30,34)(27,35,37,31)(28,32,38,40)(29,39,36,33);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,14,16,21)(12,17,19,23)(15,20,22,24)(25,27,30,37)(26,31,34,35)(28,36,38,29)(32,39,40,33);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28,30,38)(26,32,34,40)(27,36,37,29)(31,39,35,33);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,23)(22,24)(25,29,30,36)(26,33,34,39)(27,28,37,38)(31,32,35,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28)(26,32)(27,36)(29,37)(30,38)(31,39)(33,35)(34,40);(1,3)(2,5)(4,7)(6,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034313);