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 | |
4345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,17,19,13)(12,20,22,15)(18,21,24,23)(25,26,30,31)(27,32,34,29)(28,36,38,33)(35,37,40,39);(1,2)(3,8)(4,6)(5,7)(9,11,14,19)(10,13,16,17)(12,18,22,24)(15,21,20,23)(26,31)(27,34)(33,36)(35,40);(1,3,5,6)(2,4,7,8)(9,10)(11,13)(12,15)(14,16)(17,19)(18,21)(20,22)(23,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,2)(3,4)(5,7)(6,8)(9,10)(11,17)(12,15)(13,19)(14,16)(18,23)(20,22)(21,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,12,14,22)(10,15,16,20)(11,18,19,24)(13,21,17,23);(1,4)(2,6)(3,7)(5,8)(9,13,14,17)(10,11,16,19)(12,21,22,23)(15,18,20,24)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8331835);