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 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,24)(22,23);(1,3)(2,5)(4,8)(6,7)(10,15)(11,20)(12,18)(22,24)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,39)(38,40);(1,3,4,8)(2,6,7,5)(11,20)(12,18)(13,17)(16,19)(26,30)(27,34)(28,37)(39,40);(3,8)(5,6)(9,11,14,20)(10,16,15,19)(12,23,18,21)(13,22,17,24)(25,27)(26,31)(28,38)(29,34)(30,33)(32,40)(35,37)(36,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,18)(10,17,15,13)(11,21,20,23)(16,22,19,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,39)(33,40)(34,38);(1,2)(3,5)(4,7)(6,8)(9,13,14,17)(10,18,15,12)(11,22,20,24)(16,21,19,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965448);