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 | #38509 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,21)(13,18)(14,16)(17,22)(19,23)(20,24)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(2,3)(6,7)(9,11,14,19)(10,15,16,23)(12,20,22,18)(13,21,24,17)(25,27,31,36)(26,32,35,38)(28,30,37,39)(29,33,34,40);(1,3,5,2)(4,7,8,6)(10,16)(11,19)(17,21)(18,20)(27,36)(28,37)(32,38)(33,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,20)(13,17)(14,19)(16,23)(18,22)(21,24)(25,28)(26,33)(27,30)(29,38)(31,37)(32,34)(35,40)(36,39);(9,12,14,22)(10,17,16,21)(11,20,19,18)(13,23,24,15)(25,29)(26,30)(27,33)(28,38)(31,34)(32,37)(35,39)(36,40);(9,13)(10,18)(11,21)(12,15)(14,24)(16,20)(17,19)(22,23)(25,30)(26,34)(27,37)(28,36)(29,35)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8619640);