Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,15,16,12)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(2,3)(6,7)(10,14)(11,16)(17,18,21,20)(19,23,24,22)(27,34)(28,36)(30,33)(31,38);(4,8)(6,7)(9,11,13,16)(10,15,14,12)(17,19,21,24)(18,22,20,23)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39);(1,3,5,2)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18,21,20)(19,22,24,23)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39);(1,4)(2,6)(3,7)(5,8)(9,12,13,15)(10,11,14,16)(17,20,21,18)(19,23,24,22)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=gh,d^2=fh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3304146);