Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18,20,22)(19,21,23,24)(26,30)(31,33)(32,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,16,12)(19,23)(21,24)(28,36)(32,37)(34,40)(38,39);(1,2,4,5)(3,7,8,6)(9,10,13,16)(11,14,15,12)(17,19)(18,21)(20,23)(22,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,15,16,11)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393688);