Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| 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 |
| prime radical | #16508 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,14)(11,16)(17,18,21,22)(19,23,24,20)(25,26,30,29)(27,32,31,28);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,20)(21,22)(23,24)(25,26,30,29)(27,28,31,32);(2,6)(5,7)(10,14)(12,15)(18,22)(20,23)(25,27)(26,28)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,15,14,12)(17,19)(18,20)(21,24)(22,23)(25,28,30,32)(26,31,29,27);(1,2)(3,5)(4,6)(7,8)(9,12,13,15)(10,16,14,11)(17,20)(18,19)(21,23)(22,24)(25,29,30,26)(27,32,31,28)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4321975);