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 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,15)(13,14)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(10,13)(11,15)(18,21)(22,24)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(9,11,14,15)(10,12,13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,15,13,11)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(1,3,4,7)(2,5,6,8)(9,13,14,10)(11,12,15,16)(17,19,20,23)(18,22,21,24)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=g,d^2=g,e^2=fghi,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, 4185576);