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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,22,20,23)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,39,38,40);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,15)(13,16)(26,30)(28,37)(31,33)(34,38);(2,5)(6,7)(9,11,13,16)(10,15,12,14)(17,19,21,20)(18,22,24,23)(25,27,29,35)(26,31,30,33)(28,38,37,34)(32,40,36,39);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,14,16,15)(17,20,21,19)(18,23,24,22)(25,28,29,37)(26,32,30,36)(27,34,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=fgh,c^2=fghi,d^2=fgh,e^2=fghi,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, 5380921);