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 | #24364 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1719 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)(9,10,13,15)(11,14,12,16);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,22)(21,25)(24,28)(26,29)(27,31)(30,32);(9,12,13,11)(10,14,15,16)(18,22)(20,28)(23,29)(24,25)(26,32)(30,31);(2,6)(5,7)(17,19,21,27)(18,23,25,31)(20,26,28,32)(22,29,24,30);(3,8)(5,7)(18,22)(20,28)(23,29)(24,25)(26,32)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,d^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, 6599942);