Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1480 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,21,25)(19,24,20,23)(22,26,29,32)(27,31,28,30);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32);(2,6)(5,8)(10,13)(14,15)(17,19,21,20)(18,23,25,24)(22,28,29,27)(26,31,32,30);(9,11,12,16)(10,14,13,15);(17,20,21,19)(18,24,25,23)(22,27,29,28)(26,30,32,31)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6586951);