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 | |
1504 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)(9,10,13,16)(11,12,15,14)(25,26)(27,35)(28,33)(29,39)(30,31)(32,36)(34,40)(37,38);(2,3)(6,7)(9,11,13,15)(10,14,16,12)(18,21)(22,24)(26,31)(27,36)(33,37)(34,39);(1,2,5,3)(4,7,8,6)(9,10,13,16)(11,14,15,12)(17,18)(19,22)(20,21)(23,24)(25,27)(26,32)(28,34)(29,33)(30,36)(31,35)(37,40)(38,39);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,19,20,23)(18,22,21,24)(25,28)(26,33)(27,34)(29,32)(30,38)(31,37)(35,40)(36,39);(1,4,5,8)(2,6,3,7)(9,12,13,14)(10,15,16,11)(25,29)(26,34)(27,37)(28,35)(30,40)(31,39)(32,38)(33,36)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fg,d^2=gh,e^2=fg,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, 3281822);