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 | |
4548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,15,18,12)(13,21,22,17)(19,20,24,23)(25,26,31,30)(27,34,32,28)(29,37,38,33)(35,36,39,40);(2,3)(6,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,21)(22,24)(25,27,31,32)(26,28,30,34)(29,35,38,39)(33,36,37,40);(1,3,5,2)(4,7,8,6)(10,16)(11,18)(17,21)(19,24)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,39,37,35);(1,4)(2,6)(3,7)(5,8)(9,12,14,15)(10,11,16,18)(13,20,22,23)(17,19,21,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(9,13,14,22)(10,17,16,21)(11,19,18,24)(12,20,15,23)(25,29,31,38)(26,33,30,37)(27,35,32,39)(28,36,34,40);(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,21)(22,24)(25,30,31,26)(27,34,32,28)(29,37,38,33)(35,40,39,36)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=g,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104234);