Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
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 | |
1674 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)(9,10,11,14,12,13,15,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(3,8)(5,7)(17,19,22,29)(18,23,25,26)(20,28,21,27)(24,32,30,31);(1,3,4,8)(2,5,6,7)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32);(2,6)(5,7)(17,21,22,20)(18,24,25,30)(19,28,29,27)(23,31,26,32);(10,13)(14,16)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfg,d^b=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314845);