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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,12,17,16)(13,18,23,24)(19,20,21,22)(25,26,32,31)(27,34,28,33)(29,36,30,35)(37,39,38,40);(1,2)(3,7)(4,8)(5,6)(10,15)(11,17)(18,24)(19,21)(25,27,32,28)(26,33,31,34)(29,37,30,38)(35,39,36,40);(1,3)(2,4)(5,8)(6,7)(25,28,32,27)(26,34,31,33)(29,38,30,37)(35,40,36,39);(2,6)(4,7)(9,11,14,17)(10,16,15,12)(13,21,23,19)(18,20,24,22)(25,29,32,30)(26,35,31,36)(27,37,28,38)(33,39,34,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,24,21)(25,30,32,29)(26,36,31,35)(27,38,28,37)(33,40,34,39);(1,4,5,7)(2,8,6,3)(9,13,14,23)(10,18,15,24)(11,19,17,21)(12,20,16,22)(25,31,32,26)(27,34,28,33)(29,35,30,36)(37,39,38,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=g,f^2=gi,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, 8606196);