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 | |
4577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,19)(12,22)(13,18)(14,15)(16,20)(17,23)(21,24)(26,30)(31,33)(32,37)(38,39);(2,5)(6,7)(10,15)(11,20)(17,22)(18,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,24)(16,19)(17,22)(28,36)(32,37)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,4,8)(2,6,5,7)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=i,d^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^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8241604);