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 | |
4447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(26,32)(27,36)(28,34)(30,39);(1,2,4,6)(3,5,8,7)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(25,26,31,32)(27,33,36,40)(28,29,34,37)(30,35,39,38);(1,2)(3,5)(4,6)(7,8)(10,16)(12,17)(15,22)(18,23)(25,27,31,36)(26,33,32,40)(28,38,34,35)(29,30,37,39);(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,28,31,34)(26,29,32,37)(27,35,36,38)(30,33,39,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29)(26,34)(27,30)(28,32)(31,37)(33,35)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,30)(26,35)(27,37)(28,33)(29,36)(31,39)(32,38)(34,40)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9694369);