Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,13,12)(11,14,15,16)(19,23)(21,24)(26,30)(28,36)(31,33)(34,40);(2,6)(4,7)(11,15)(14,16)(17,18,20,22)(19,21,23,24)(28,36)(32,37)(34,40)(38,39);(1,3,5,8)(2,4,6,7)(9,11,13,15)(10,14,12,16)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,7)(4,6)(5,8)(17,19)(18,21)(20,23)(22,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,16,15,14)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4086902);