Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1810 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)(3,7)(4,5)(6,8)(11,15)(14,16)(17,18,20,21)(19,23,24,22)(26,27)(28,32);(1,3)(2,6)(4,7)(5,8)(9,10,13,12)(11,14,15,16)(17,18)(19,22)(20,21)(23,24)(26,27)(30,31);(2,4)(6,7)(18,21)(22,23)(25,26,29,27)(28,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,29,26)(28,31,32,30);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,16,15,14)(17,18)(19,22)(20,21)(23,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=fg,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2409958);