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 | |
4408 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(26,31)(27,36)(34,38)(35,40);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,26)(27,32)(28,34)(29,35)(30,31)(33,36)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27)(26,32)(28,35)(29,34)(30,36)(31,33)(37,40)(38,39);(1,3,4,8)(2,5,6,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,27)(26,33)(28,35)(29,38)(30,36)(31,32)(34,39)(37,40);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,28,30,37)(26,34,31,38)(27,35,36,40)(29,33,39,32);(9,10)(11,15)(12,16)(13,22)(14,17)(18,21)(19,23)(20,24)(25,29,30,39)(26,35,31,40)(27,34,36,38)(28,33,37,32)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8345022);