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 | #47955 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,15,20,13)(12,16,22,24)(18,23,19,21)(25,26,31,34)(27,35,38,32)(28,30,36,40)(29,33,39,37);(2,3)(6,7)(9,11,14,20)(10,15,17,13)(12,19,22,18)(16,21,24,23);(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,27,31,38)(26,32,34,35)(28,29,36,39)(30,37,40,33);(2,3)(4,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,21)(17,24)(18,20)(25,28)(26,30)(27,29)(31,36)(32,37)(33,35)(34,40)(38,39);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,18)(14,20)(15,17)(16,21)(19,22)(23,24)(25,29)(26,33)(27,36)(28,38)(30,32)(31,39)(34,37)(35,40);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,21)(14,15)(16,18)(17,20)(19,24)(22,23)(25,30)(26,28)(27,37)(29,32)(31,40)(33,38)(34,36)(35,39)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10326821);