Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1352 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)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,20,21,25,31,30)(19,24,28,29,26,23,32,27);(1,3,4,8)(2,5,6,7)(17,19,21,26)(18,23,25,24)(20,27,30,29)(22,28,31,32);(3,8)(5,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,29)(21,26)(22,28)(23,25)(27,30)(31,32);(2,6)(5,7)(18,25)(19,26)(20,30)(28,32);(9,12,13,11)(10,14,15,16)(17,20,22,25,21,30,31,18)(19,27,28,24,26,29,32,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286169);