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 | |
4678 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)(3,4)(5,7)(6,8)(10,14)(11,17)(16,21)(18,23)(25,26,30,34)(27,32,28,31)(29,33,38,37)(35,40,36,39);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(2,7)(3,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,40)(36,39);(1,4,5,6)(2,3,7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,27)(26,32,34,31)(29,35,38,36)(33,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8603922);