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 | |
4483 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)(3,7)(4,5)(6,8)(9,10,14,13)(11,15,18,16)(12,17,20,21)(19,22,24,23)(25,26,31,33)(27,36,30,32)(28,34,38,40)(29,37,39,35);(2,5)(6,7)(9,11)(10,15)(12,19)(13,16)(14,18)(17,22)(20,24)(21,23)(25,27,31,30)(26,32,33,36)(28,35,38,37)(29,34,39,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,16)(19,24)(22,23)(26,33)(28,38)(32,36)(35,37);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,28,31,38)(26,34,33,40)(27,37,30,35)(29,36,39,32);(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,29)(26,35)(27,34)(28,36)(30,40)(31,39)(32,38)(33,37);(9,13,14,10)(11,16,18,15)(12,17,20,21)(19,22,24,23)(25,30,31,27)(26,32,33,36)(28,37,38,35)(29,34,39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8654120);