Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| 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 | |
1440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,13,12)(11,14,15,16)(19,23)(20,21)(25,26)(27,30)(28,33)(29,34)(31,36)(32,37)(35,38)(39,40);(11,15)(14,16)(17,18,22,24)(19,20,23,21)(26,27)(29,37)(31,33)(35,40);(1,2)(3,7)(4,8)(5,6)(9,11,13,15)(10,14,12,16)(17,19)(18,20)(21,24)(22,23)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3,5,8)(2,7,6,4)(17,20,22,21)(18,23,24,19)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,16,15,14)(17,21,22,20)(18,19,24,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=h,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1033832);