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 | #43043 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,4,5)(11,17)(15,18)(19,23)(22,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,3)(2,7)(4,6)(5,8)(9,10,14,12)(11,18,17,15)(13,16,21,20)(19,24,23,22)(25,27)(26,31)(28,30)(29,35)(32,39)(33,34)(36,38)(37,40);(2,8)(3,4)(10,12)(15,18)(16,20)(22,24)(25,26,30,33)(27,31,28,34)(29,32,38,37)(35,39,36,40);(1,3,6,4)(2,5,8,7)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,4,6,3)(2,5,8,7)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,5)(2,4)(3,8)(6,7)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9976153);