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 | |
4339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26,31,28)(27,29,32,34)(30,33,38,36)(35,37,39,40);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(11,17)(14,21)(18,23)(22,24)(25,26,31,28)(27,34,32,29)(30,33,38,36)(35,40,39,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,28,31,26)(27,34,32,29)(30,33,38,36)(35,37,39,40);(1,3,4,7)(2,6,5,8)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,29,31,34)(26,32,28,27)(30,37,38,40)(33,39,36,35);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30,31,38)(26,33,28,36)(27,35,32,39)(29,37,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=hi,d^2=hi,e^2=ghi,f^2=gi,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, 8488722);