Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,13,14)(11,12,16,15)(17,18,21,22)(19,24,20,23)(25,26)(27,30)(28,32)(29,31);(2,5)(6,8)(9,10,13,14)(11,15,16,12)(18,22)(23,24)(25,27,29,32)(26,28,31,30);(10,14)(11,16)(17,19,21,20)(18,23,22,24)(25,27,29,32)(26,30,31,28);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,21,19)(18,24,22,23)(25,27)(26,28)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=f,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5872019);