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 | |
4392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,21)(16,19)(22,23)(25,26)(27,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(27,34)(30,31)(35,40)(38,39);(1,3,5,4)(2,6,7,8)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40)Polycyclic presentation:
2^9|c^2=hi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8336547);