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 | |
4265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,16)(12,18)(13,17)(14,19)(15,22)(20,23)(21,24)(26,30)(27,33)(32,37)(34,39);(1,2,5,3)(4,6,8,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,23)(18,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(2,3)(6,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,24)(18,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,12,14,13)(10,17,19,18)(11,20,22,21)(15,23,16,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24);(1,4)(2,6)(3,7)(5,8)(9,13,14,12)(10,18,19,17)(11,21,22,20)(15,24,16,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|b^2=hi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8001935);