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 | |
4476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,16,19)(11,20,22,14)(12,17,13,23)(15,18,21,24)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(2,5)(6,7)(9,11)(10,14)(12,21)(13,15)(16,22)(17,18)(19,20)(23,24)(27,33)(30,36)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,17,19,23)(11,21,22,15)(14,18,20,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13,16,12)(10,17,19,23)(11,21,22,15)(14,24,20,18)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,11)(12,18)(13,24)(15,23)(16,20)(17,21)(19,22)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(9,15,16,21)(10,18,19,24)(11,13,22,12)(14,17,20,23)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=gi,d^2=gh,f^2=gh,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, 8603995);