Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631112 |
| 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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1482 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)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,26,29)(20,24,28,30)(25,31,27,32);(1,3,4,8)(2,5,6,7)(18,22)(23,29)(24,30)(31,32);(1,2)(3,7)(4,6)(5,8)(10,13)(11,16)(14,15)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31);(2,6)(5,7)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32);(10,14)(13,15)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^c=dfh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538787);