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 | |
4380 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)(3,6)(4,7)(5,8)(9,10,14,15)(11,16,19,23)(12,17,22,21)(13,20,24,18)(25,26)(27,28)(29,37)(30,31)(32,34)(33,39)(35,38)(36,40);(1,3)(2,5)(4,8)(6,7)(10,15)(12,22)(16,23)(18,20)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,38)(39,40);(10,15)(11,19)(17,21)(18,20)(26,31)(27,34)(33,37)(35,40);(9,11)(10,16)(12,20)(13,21)(14,19)(15,23)(17,24)(18,22)(25,27)(26,32)(28,31)(29,35)(30,34)(33,38)(36,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,20)(13,23)(14,22)(15,21)(16,24)(18,19)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,40,37,35);(1,2)(3,5)(4,7)(6,8)(9,13,14,24)(10,18,15,20)(11,21,19,17)(12,16,22,23)(25,29,30,39)(26,33,31,37)(27,35,34,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9625742);