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 |
| prime radical | #36396 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(10,15)(11,17)(12,19)(21,24)(25,26,31,35)(27,32,36,40)(28,30,34,39)(29,38,37,33);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,13,17,22)(12,20,19,16)(18,24,23,21)(27,36)(29,37)(32,40)(33,38);(1,2)(3,6)(4,5)(7,8)(25,27,31,36)(26,32,35,40)(28,37,34,29)(30,33,39,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28)(26,30)(27,37)(29,36)(31,34)(32,33)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12,14,19)(10,16,15,20)(11,18,17,23)(13,21,22,24)(25,29,31,37)(26,33,35,38)(27,28,36,34)(30,32,39,40);(9,13)(10,17)(11,15)(12,21)(14,22)(16,23)(18,20)(19,24)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=hi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400283);