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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4686 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)(3,4,7,6)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,32)(28,29)(30,37)(31,35)(33,40)(34,36)(38,39);(1,2)(3,6)(4,7)(5,8)(10,14)(15,17)(16,20)(22,23)(25,27,31,38)(26,32,35,39)(28,37,40,34)(29,30,33,36);(3,7)(4,6)(10,14)(11,18)(16,20)(19,24)(25,28,31,40)(26,33,35,29)(27,34,38,37)(30,39,36,32);(1,3,5,7)(2,6,8,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,29,31,33)(26,28,35,40)(27,36,38,30)(32,34,39,37);(1,4,5,6)(2,7,8,3)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28)(26,33)(27,37)(29,35)(30,39)(31,40)(32,36)(34,38);(1,2)(3,6)(4,7)(5,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30,31,36)(26,34,35,37)(27,33,38,29)(28,39,40,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=gh,e^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, 8034826);