Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, cosocle | C24 | C25 |
| prime radical | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,16,12)(17,18)(19,22)(20,21)(23,24)(25,26,29,32)(27,28,31,30);(2,3)(4,8)(9,11,13,16)(10,12,14,15)(18,20)(22,23)(25,26)(27,28)(29,32)(30,31);(2,3)(6,7)(10,14)(12,15)(17,19,21,24)(18,22,20,23)(25,27,29,31)(26,30,32,28);(1,3,5,2)(4,6,8,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,22)(21,24)(25,27)(26,30)(28,32)(29,31);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,21,18)(19,23,24,22)(25,28,29,30)(26,31,32,27)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=fgh,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3075528);