Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,12,15)(19,20)(23,24)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,37,39);(1,3)(2,6)(4,7)(5,8)(18,22)(23,24)(25,27,30,32)(26,31,34,28)(29,35,38,37)(33,39,40,36);(1,3,5,8)(2,7,4,6)(9,11,13,12)(10,14,16,15)(17,18,21,22)(19,23,20,24)(25,27)(26,28)(29,37)(30,32)(31,34)(33,39)(35,38)(36,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,20)(18,23,22,24)(25,28)(26,32)(27,34)(29,36)(30,31)(33,37)(35,40)(38,39);(1,4,5,2)(3,6,8,7)(9,12,13,11)(10,15,16,14)(17,20,21,19)(18,24,22,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,d^2=fgh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5357669);