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 | |
4495 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)(11,19)(12,17)(13,20)(14,16)(15,22)(18,24)(21,23)(25,26,31,33)(27,32,36,40)(28,29,34,38)(30,37,39,35);(1,2)(3,7)(4,5)(6,8)(9,11,14,22)(10,15,16,19)(12,20,23,18)(13,17,24,21)(25,27,31,36)(26,32,33,40)(28,35,34,37)(29,30,38,39);(2,5)(6,7)(10,16)(12,23)(15,19)(18,20)(26,33)(27,36)(29,38)(35,37);(2,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,28,31,34)(26,29,33,38)(27,37,36,35)(30,32,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,23)(10,17,16,21)(11,20,22,18)(13,15,24,19)(25,29)(26,34)(27,30)(28,33)(31,38)(32,37)(35,40)(36,39);(1,3,4,8)(2,6,5,7)(9,13,14,24)(10,18,16,20)(11,21,22,17)(12,15,23,19)(25,30)(26,35)(27,29)(28,32)(31,39)(33,37)(34,40)(36,38)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10304483);