Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2433 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18,20,24)(19,23,27,22)(21,25,28,32)(26,31,29,30);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29);(9,12,13,11)(10,14,15,16)(17,19,20,27)(18,22,24,23)(21,29,28,26)(25,31,32,30);(2,6)(5,7);(3,8)(5,7)(17,19,20,27)(18,23,24,22)(21,29,28,26)(25,30,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6599947);