Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2098 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)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,18)(19,25)(20,31)(21,27)(22,26)(23,28)(24,29)(30,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,19,23,30)(18,24,28,31)(20,21,29,22)(25,27,32,26);(1,4,5,3)(2,8,6,7)(17,20)(18,25)(19,21)(22,30)(23,29)(24,27)(26,31)(28,32);(17,21,23,22)(18,26,28,27)(19,29,30,20)(24,25,31,32)Polycyclic presentation:
2^9|b^2=g,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354219);