Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,11)(12,16)(17,18,21,22)(19,20,24,23)(27,31)(30,32);(1,3,5,8)(2,6,7,4)(10,11)(14,15)(18,22)(19,24)(26,29)(30,32);(2,7)(3,8)(9,10,13,11)(12,14,16,15)(17,19,21,24)(18,23,22,20)(25,26)(27,30)(28,29)(31,32);(1,2)(3,4)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,24)(22,23);(1,4)(2,3)(5,6)(7,8)(9,12,13,16)(10,14,11,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=fi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363436);